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ideal generated by the λ summand is exactly the sum of the µ summands over all µ with λ ⊂ µ. (Here ⊂ denotes containment of Young diagrams.) Another theorem along these lines, that we proved [NSS1], concerns finiteness properties of equivariant modules that are uniform in V and W : one can regard (V, W ) 7→ Sym(V ⊗ W ) as an algebra object in the category of bivariate polynomial functors, and we show that it is noetherian. (Omitting duality does not change much, and makes the functorial properties nicer.) The proof of this theorem crucially relies upon the aforementioned result of [CEP]. Similar pictures are known in some related contexts. For example, consider the space Sym2(V ∗) of symmetric bilinear forms on V . The coordinate ring Sym(Sym2(V )) is again multiplicity free as a representation of GL(V ). The equivariant ideals were classified in [Ab], and the noetherian result was established in [NSS1]. See [AdF, NSS2, SS2] for some other cases. The purpose of this paper is to establish analogs of the above results in a new situation. An isomeric is a super vector space V equipped with an odd-degree α: V → V squaring to the identity. The isomeric supergroup (also known as the queer supergroup) Q(V ) is the automorphism supergroup of (V,α). Given two isomeric vector spaces (V,α) and (W, β), one can consider the supervariety X of all linear maps V → W that are compatible with the isomeric structure. The supergroup Q(V ) × Q(W ) acts on X. We classify the equivariant ideals of C[X] and establish a noetherian result. 1.2. Statement of results. We now state our main results in detail. Let q be the infinite isomeric Lie (see §2.4). There is a notion of polynomial representation for q, analogous to that for the general linear group (see §2.5). A (bivariate) isomeric algebra is a supercommutative superalgebra equipped with an action of q × q under which it forms a polynomial representation. Let V be the standard representation of q and let U be the half tensor product 2−1(V ⊗ V), which is a polynomial representation of q × q. (See §2.3 for the definition of half tensor product.) Let A = Sym(U). This is an isomeric algebra, and the main one of interest in this paper. The space Spec(A) is (more or less) the space of infinite isomeric matrices. The irreducible polynomial representations of q are parametrized by strict partitions. Let Tλ be the irreducible representation corresponding to the strict partition λ. An analog of the Cauchy decomposition yields the decomposition −δ(λ) (1.1) A = 2 (Tλ ⊗ Tλ), where the sum is over all strict partitionsM λ, δ(λ) is either 0 or 1, and a 2−1 indicates a half tensor product. Each summand here is an irreducible representation of q × q. This shows that A is multiplicity free as a representation of q × q. In particular, we see that an equivariant ideal of A (meaning one stable by q × q) is determined by which summands it contains. Our first main result is the following, which completely classifies the equivariant ideals of A: Theorem 1.2. Let Iλ be the ideal of A generated by the λ summand in (1.1). Then Iλ contains the µ summand if and only if λ ⊆ µ. The key idea in proving Theorem 1.2 is to use Schur–Sergeev duality to convert to a more combinatorial problem; see §3.1 for a more detailed outline. We note that this theorem is the isomeric analog of the aforesaid result of [CEP]. As a simple application of this theorem, we describe the isomeric determinantal ideals (see Remark 3.20). ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 3

Theorem 1.2 implies that the lattice of ideals of A is isomorphic to the lattice of ideals in the poset of strict partitions. From this, it follows that equivariant ideals of A satisfy the ascending chain condition. This is a noetherian-like result; however, in practice one wants a much stronger statement. Let ModA be the category of q × q equivariant A-modules that form a polynomial repre- sentation of q × q. In other words, treating A as an algebra object of the tensor category pol Rep (q × q), the category ModA is the category of A- objects. We say that A is noetherian (as an isomeric algebra) if the category ModA is locally noetherian. Equiva- lently, this means that any subobject of a finitely generated object of ModA is again finitely generated. The corollary of Theorem 1.2 observed above shows that any subobject of A itself is finitely generated. This is not enough to conclude that A is noetherian: a general finitely generated A-module is a quotient of V ⊗ A for some finite length polynomial representation V , but typically not a quotient of a finite of A’s. Our second main theorem yields the desired strengthening: Theorem 1.3. The bivariate isomeric algebra A is noetherian. The proof of Theorem 1.3 follows the general plan employed in previous noetherian results we have proved [NSS1, NSS2, SS2]. The basic idea is to break ModA up into two pieces: the tors gen torsion subcategory ModA and the generic category ModA (which is the Serre quotient by tors the torsion subcategory). Thanks to Theorem 1.2, one can show that ModA is locally noe- gen therian. The crucial step is to identify ModA with the category of algebraic representations gen of q as studied in [GS]. We thus see that ModA inherits the pleasant properties of this rep- resentation category (such as: all objects are locally finite length, and finite length objects tors have finite injective ). We then piece together the information about ModA and gen ModA to obtain our result on ModA. We note that our proof of Theorem 1.3 yields quite a bit of extra useful information. For gen example, as stated above, we determine the structure of the generic category ModA . We gen also prove some important results about the section functor S : ModA → ModA. Some further results can be easily deduced: for instance, projective A-modules are injective. Remark 1.4. The simplest example of an isomeric algebra is Sym(V). It is easily seen to be noetherian, see Remark 2.3. The algebra A is, in a sense, the smallest isomeric algebra where the noetherian property is not obvious.  1.3. Motivation. In [SS3], the second and third authors investigate the Brauer category and its relatives (this is actually the first in a series of papers on this topic). One relative is the “isomeric walled Brauer category,” which relates to mixed tensor representations of the isomeric algebra. Theorem 1.3 implies a noetherian result for this category that will be important for that project. This was our motivation for studying A specifically. We also have a more general motivation for the current project. A GL-algebra is a commutative algebra equipped with an action of the infinite general linear group GL under which it forms a polynomial representation. In characteristic 0, these are equivalent to twisted commutative algebras (tca’s) by Schur–Weyl duality. In recent years, GL-algebras and tca’s have received a lot of attention. All indications so far are that these are very well-behaved objects, though there is much that is still unknown. The isomeric algebras studied in this paper are close relatives of GL-algebras. Given that GL-algebras are such a rich topic, we expect that isomeric algebras are as well. This paper represents the first attempt to study them in any serious capacity. 4 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

1.4. Future work. As mentioned, it seems that isomeric algebras are likely just as rich as GL-algebras. Developing the theory of isomeric algebras in more detail is therefore a potentially fruitful direction for future work. A more specific problem is to classify the equivariant prime ideals of the isomeric algebra A studied in this paper. The third author tackled the analogous problem for the GL-algebra Sym(Sym2(C∞)) in [Sn]. He is currently pursuing this problem with Robert Laudone, and the result will appear in the forthcoming work [LS].

1.5. A remark on terminology. The supergroup Q(n) has traditionally been called the “queer supergroup.” Since the connotations of this word have shifted over the years, we have introduced the term “isomeric” to take its place1. This word is derived from the Greek words isos, meaning equal, and m´eros, meaning part, and is intended to reflect the fact that an isomeric structure on a super vector space makes its two parts isomorphic. We hope that other authors will choose to use this new terminology as well. We thank the conscientious editor who suggested that we change the terminology.

1.6. Outline. In §2, we review isomeric vector spaces, the isomeric , and isomeric algebras. In §3, we classify the ideals of A. In §4, we show that equivariant A- modules are locally free at a generic point of Spec(A). Using this, we describe the generic gen category ModA in §5. Finally, we prove the noetherian result in §6.

1.7. Notation. We list the most important notation: ζ: a fixed square root of −1 in C V: the complex super vector space with basis {ei, fi}i≥1 α: the isomeric structure on V given by α(ei)= fi q: the isomeric algebra associated to (V,α) 2−1(− ⊗ −): the half tensor product (see §2.3) U: the half-tensor product 2−1(V ⊗ V), a representation of q × q A: the q × q-algebra Sym(U) Tλ: the isomeric Schur functor associated to the strict partition λ

2. Isomeric algebras 2.1. Super vector spaces. A super vector space is a Z/2-graded complex vector space V = V0 ⊕ V1. For a homogeneous element x ∈ V , we write |x| ∈ Z/2 for its degree. We let Cn|m be the super vector space with degree 0 part Cn and degree 1 part Cm. We let n|m e1,...,en, f1,...,fm be the standard basis for C , where the ei’s have degree 0 and the ∞|∞ n|n fi’s have degree 1. We let V = C = n≥1 C , and sometimes write W for a copy of V. Given a super vector space V and k ∈ Z/2, we let V [k] be the super vector space defined by S V [k]i = Vk+i. (We remark that V [1] is often denote Π(V ) in the literature, e.g., in [CW1].) Let V and W be super vector spaces. We let Hom(V, W ) denote the set of all linear maps V → W . We let Hom(V, W )k denote the subset of Hom(V, W ) consisting of maps f such that f(Vi) ⊂ Wi+k for all i. Then Hom(V, W ) = Hom(V, W )0 ⊕ Hom(V, W )1, and so Hom(V, W ) is naturally a super vector space.

1The word “isomeric” is not new: it occurs in chemistry as the adjectival form of “isomer.” However, it does not appear to be used within pure mathematics. ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 5

We let SVec denote the category whose objects are super vector spaces and whose maps are all linear maps. We let SVec◦ be the subcategory where the are homogeneous of even degree. Given super vector spaces V and W , let V ⊗ W be the usual tensor product of V and W , endowed with the usual grading. Given homogeneous linear maps f : V → V ′ and g : W → W ′, we let f ⊗ g be the V ⊗ W → V ′ ⊗ W ′ defined on homogeneous elements by (2.1) (f ⊗ g)(v ⊗ w)=(−1)|g||v|f(v) ⊗ g(w). This construction endows SVec◦ with a monoidal structure. This monoidal structure admits a symmetry τ defined by τ(x ⊗ y)= (−1)|x||y|y ⊗ x. We note that ⊗ does not define a monoidal structure on SVec in the usual sense, as it does not even define a functor SVec × SVec → SVec due to (2.1); however, it does define a monoidal structure on SVec in the sense of supercategories, as we now discuss.

2.2. Supercategories. A supercategory is a category enriched in super vector spaces, or, more precisely, the category SVec◦. Concretely, a supercategory is just a linear category in which there is a notion of even and odd homogeneous , satisfying some rules; most categories built out of super vector spaces will therefore be supercategories. We refer to [SS4, §2] as a general reference, though this concept has been discussed elsewhere as well (e.g., [BE]). Given a supercategory C, we let C◦ be the ordinary category with the same objects and with HomC◦ (X,Y ) = HomC(X,Y )0. There is a natural notion of product of supercategories, which leads to the notion of a monoidal supercategory; see [SS4, §3.1] or [BE, Definition 1.4]. If (C, ⊗) is a monoidal supercategory then ⊗ induces a monoidal operation on C◦. The monoidal product ⊗ is defined on all pairs of morphisms, and satisfies the sign rule (2.1). The notion of a symmetry on a monoidal supercategory is defined in the usual manner. (We note that the paper [SS4] is devoted to studying the notion of a on a monoidal supercategory, which is more complicated than the notion of a symmetry. We will not need in this paper.) The basic example of a monoidal supercategory is SVec with the monoidal operation ⊗ discussed above; τ defines a symmetry on this monoidal structure.

2.3. Isomeric vector spaces. A isomeric vector space is a pair (V,α) consisting of a super vector space V together with an odd endomorphism α: V → V such that α2 = 1. We n|n endow C with an isomeric structure by α(ei)= fi and α(fi)= ei; we call this the standard isomeric structure. We similarly define the standard isomeric structure on V = C∞|∞. Every finite dimensional isomeric vector space is isomorphic to some Cn|n equipped with the standard isomeric structure. A homogeneous morphism f : (V,α) → (W, β) of isomeric vector spaces is a homogeneous linear map f : V → W such that fα =(−1)|f|βf; a general morphism is a sum of homogeneous morphisms. We let QVec denote the supercategory of isomeric vector spaces. Let (V,α) and (W, β) be two isomeric vector spaces. Then α ⊗ β is an even endomorphism of V ⊗ W that squares to −1; the sign comes from (2.1). Fix once and for all a square root of −1 in C, denoted ζ. We define the half tensor product of (V,α) and (W, β), denoted 2−1(V ⊗ W ), to be the ζ eigenspace of α ⊗ β. We note that α ⊗ 1 induces an isomorphism between the ζ and −ζ eigenspaces, and so we have a natural isomorphism V ⊗ W ∼= 2−1(V ⊗ W ) ⊗ C1|1. Abstractly, the half tensor product can be understood using 6 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

the theory of supersymmetric monoidal categories developed in [SS4], but this perspective will not be needed in this paper.

2.4. The isomeric Lie superalgebra. Let (V,α) be a finite dimensional isomeric vector space. The isomeric Lie superalgebra q(V ) = q(V,α) is the set of endomorphisms of (V,α) in the category QVec. It forms a sub Lie superalgebra of the general linear Lie n|n superalgebra gl(V ). We let qn = q(C ). Explicitly, an element of qn can be described as a matrix of the form a b −b a   where a and b are n × n matrices. We define the Chevalley automorphism τ of qn by a b at ζbt τ = − −b a −ζbt at     where (−)t denotes the usual matrix transpose. One verifies that this is a Lie superalgebra . Note that a b a −b τ 2 = , −b a b a    

and so τ has order four. We let q = q∞ = n≥1 qn be the infinite isomeric superalgebra. It can be represented by matrices as above, and we define τ on q as above. S

n|n 2.5. Polynomial representations. The space C is naturally a representation of qn, and called the standard representation. We say that a representation of qn is polynomial if it occurs as a subquotient of a (possibly infinite) direct sum of tensor powers of the standard pol representation. We let Rep (qn) denote the supercategory of such representations. It is easily seen to be closed under tensor product. It follows from Schur–Sergeev duality [CW1, pol ◦ §3.4] that the abelian category Rep (qn) is semi-simple. Recall that a strict partition of n is a partition λ of n such that λ1 > λ2 > ··· > λr > 0; with this notation, we write |λ| = n and ℓ(λ)= r. For each strict partition λ with ℓ(λ) ≤ n, there is a simple object Tλ,n pol ◦ of Rep (qn) , and every simple object is isomorphic to some Tλ,n or Tλ,n[1]. To be a bit more precise, define 0 if ℓ(λ) is even δ(λ)= . (1 if ℓ(λ) is odd ∼ Then there is an even isomorphism Tλ,n = Tλ,n[1] if δ(λ) = 1, and these are the only among objects of the form Tµ,n[k] with k ∈ Z/2. The above discussion applies equally well to the infinite isomeric algebra q. Precisely, we define a representation of q to be polynomial if it occurs as a subquotient of a direct sum of tensor powers of the standard representation V. The category Reppol(q)◦ is again semi-simple and closed under tensor products. Given a strict partition λ, there is a simple object Tλ, and every simple object is isomorphic to some Tλ or Tλ[1]. We define the notion of polynomial representation of q × q in an analogous manner. Similar results hold for it (e.g., its simple objects have the form Tλ ⊗ Tµ or Tλ ⊗ Tµ[1] with λ and µ strict partitions). ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 7

2.6. Polynomial functors. Consider the supercategory Fun(QVec, SVec) of all functors QVec → SVec. The subcategory Fun(QVec, SVec)◦ consisting of even degree morphisms ⊗n is abelian. Define a functor Tn : QVec → SVec by Tn(V,α) = V . By [CW1, Theorem 3.49], this functor is semisimple, and decomposes in Fun(QVec, SVec)◦ as a direct sum of simple functors Tλ (and their shifts) indexed by strict partitions λ of n. We say that a functor QVec → SVec is polynomial if it is a subquotient of a (possibly infinite) direct ◦ sum of the functors Tn in the category Fun(QVec, SVec) . We denote by QPol the full subcategory of Fun(QVec, SVec) spanned by polynomial functors. The category QPol◦ is a semisimple abelian category, and every simple object is isomorphic to some Tλ or Tλ[1]. The tensor product of two polynomial functors is again a polynomial functor. pol n|n We have an evaluation functor QPol → Rep (qn) given by F 7→ F (C ). For a strict partition λ, we have

n|n Tλ,n if n ≥ ℓ(λ) Tλ(C )= . (0 otherwise pol We similarly have an evaluation functor QPol → Rep (q) that takes Tλ to Tλ. This is an equivalence of supercategories. We also have a notion of bivariate polynomial functors. Precisely, these are subquotients of ⊗m ⊗n direct sums of functors Tm,n : QVec×QVec → SVec given by Tm,n(V, W )= V ⊗W .We let QPol(2) denote the category of such functors. It has similar properties to QPol. 2.7. Isomeric algebras. An isomeric algebra is a commutative algebra object in the symmetric QPol◦ ∼= Reppol(q)◦. A module for such an algebra is a module object in the usual sense. If M is a module over the isomeric algebra B then M(V ) is a B(V )-module for all isomeric spaces V . In this article, we are mostly interested in the bivariate analogue, i.e., commutative algebras in QPol(2); we will also call them isomeric algebras and often omit the adjective “bivariate.” Let M be a polynomial functor. We define ℓ(M) to be the supremum of the ℓ(λ) over λ for which Tλ occurs as a constituent of M. We make a similar definition for polynomial representations. We say that M is bounded if ℓ(M) < ∞. The simple functor Tν appears in Tλ ⊗Tµ if and only if Pν has nonzero coefficient when the product PλPµ is expanded in the ℓ(λ) basis of Schur P-functions (the Schur Q-functions are given by Qλ =2 Pλ [Ma, III, (8.7)] and these are, up to powers of 2, the characters of Tλ [CW1, Theorem 3.51]). It follows from [Ma, III, (8.18)] that the tensor product of two bounded representations is again bounded. We extend the notion of bounded to bivariate polynomial functors in the obvious way. An isomeric algebra A is finitely generated if it is a quotient of Sym(V ) for some finite length object V of QPol (or QPol(2)), and an A-module is finitely generated if it is a quotient of A ⊗ W for some finite length object W . We say that A is noetherian if every finitely generated A-module is noetherian, i.e., its submodules satisfy the ascending chain condition. Proposition 2.2. Every finitely generated bounded isomeric algebra is noetherian. Proof. If A is bounded, then so is A ⊗ W for any finite length object W , and hence so is any finitely generated A-module M. If ℓ(M) ≤ r, then given submodules N ⊂ N ′ ⊂ M, we have N = N ′ if and only if N(Cr|r) = N ′(Cr|r), which means it suffices to check the ascending chain condition for submodules of M(Cr|r). But this is a finitely generated module over a finitely generated supercommutative algebra A(Cr|r) (in the usual sense) and hence is noetherian. The same argument applies in the bivariate case.  8 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

n Remark 2.3. The isomeric algebra Sym(V) is bounded since Sym (V) = Tn(V). It is therefore noetherian by the above proposition. 

∞|∞ 2.8. The isomeric algebra A. Recall that V = C with basis {ei, fi}i≥1. We let U = 2−1(V ⊗ V). Recall that this is the ζ-eigenspace of α ⊗ α acting on V ⊗ V. Define

vi,j = (1+ ζ)ei ⊗ ej + (1 − ζ)fi ⊗ fj

wi,j = (1+ ζ)ei ⊗ fj + (1 − ζ)fi ⊗ ej

Then {vi,j,wi,j}i,j≥1 is a basis of U, with vi,j even and wi,j odd. The algebra q × q naturally acts on V ⊗ V and carries U into itself. Let A = Sym(U). This is a polynomial superalgebra on elements {xi,j,yi,j}i≥1, where xi,j corresponds to vi,j (and has degree 0) and yi,j to wi,j (and has degree 1). This is a bivariate isomeric algebra, and the main algebra of interest in this paper. By the isomeric analog of the Cauchy decomposition ([CW2, Theorem 3.1], the infinite case follows by taking a direct limit), we have the irreducible decomposition

−δ(λ) A = 2 (Tλ ⊗ Tλ). Mλ∈Λ Here δ(λ) is defined in §2.5. In particular, we see that A is multiplicity free as a q × q- representation.

3. Classification of ideals in A 3.1. Overview. The goal of this section is to classify the ideals in the bivariate isomeric algebra A that are stable by q × q. We now explain the basic plan of attack (notation and definitions are explained below). pol • Schur–Sergeev duality gives an equivalence of categories ModC = Rep (q), where C is a category built out of the Hecke–Clifford algebras. In fact, this is an equivalence of pol symmetric monoidal categories. There is a similar equivalence ModC⊠C = Rep (q × q). It follows that A = Sym(U) corresponds to the C ⊠ C algebra R′ = Sym(U), where U corresponds to U. Thus it suffices to classify ideals in R′. • There is a natural equivalence Cop ∼= C of symmetric monoidal categories that we call transpose. Let R be the Cop ⊠ C algebra obtained by applying transpose to the first factor in R′. Then it suffices to classify the ideals of R. op • For any category D, there is a canonical D ⊠D module given by (x, y) 7→ HomD(x, y). It turns out that R is this canonical module for Cop ⊠C. By general reasons, it follows that ideals of R correspond to tensor ideals of C. It thus suffices to classify these. • We establish a general classification of tensor ideals in monoidal supercategories of a particular form (that encompasses C). One of the key assumptions is a certain form of the Pieri rule. This section essentially follows the above plan in reverse. We thus begin with an abstract classification of tensor ideals and work our way back to A.

3.2. Semisimple . Let B be a finite dimensional semisimple C-superalgebra. ◦ By “semisimple,” we mean that the abelian category ModB is semisimple. Let Λ(B) be the set of isomorphism classes of left B-modules, where we allow odd isomorphisms. For ◦ λ ∈ Λ(B), we let Sλ be a representative simple module. Thus every simple object of ModB ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 9

is isomorphic to some Sλ or Sλ[1]. We let K(B) be the Grothendieck group of the subcate- ◦ gory of ModB spanned by finite length modules, and we let K(B) be the quotient of K(B) by the relations [M] = [M[1]], i.e., a modulee and its shift represent the same class in K(B). (The construction K(−) is discussed in [SS4, §2.6], where it is denoted K+(−).) Then K(e B) λ is the free abelian group on the classes [Sλ] with λ ∈ Λ(B). We let J ⊂ B be the Sλ-isotypic piece of B, i.e., the sum of all left ideals of B isomorphic to Sλ or Sλ[1]. Proposition 3.1. We have the following: (a) J λ is a 2-sided ideal of B. (b) J λ is simple as a (B, B)-bimodule. λ (c) B = λ∈Λ(B) J as a (B, B)-bimodule. Proof. TheseL results are proven for classical semisimple algebras in [Lan, XVII.4, XVII.5]. We just note that the proofs carry over to the super case as well.  Proposition 3.2. Let J be a left ideal of B and let J ′ be the 2-sided ideal it generates. Then ′ λ J = λ∈S J where S is the set of λ ∈ Λ(B) for which Sλ or Sλ[1] appears as a simple constituent of J. L Proof. This follows from the fact that a semisimple superalgebra B is isomorphic to a direct product of the endomorphism algebras of its simple modules and that each endomorphism algebra, as a left module, is isomorphic to a direct sum of copies of the corresponding simple (see [CW1, §3.1]). 

3.3. A classification of tensor ideals. The goal of this section is to classify left tensor ideals in certain monoidal supercategories. We first recall the relevant definitions. Definition 3.3. Let C be a supercategory. An ideal of C is a rule I assigning to each pair of objects (x, y) a homogeneous subspace I(x, y) of HomC(x, y) such that the following conditions hold: • Given f ∈ I(x, y) and g ∈ HomC(y, z), we have g ◦ f ∈ I(x, z). • Given f ∈ I(x, y) and h ∈ HomC(w, x), we have f ◦ h ∈ I(w,y). We note that I(x, x) is a 2-sided homogeneous ideal of the superalgebra EndC(x) for every object x.  Definition 3.4. Let (C, ∐) be a monoidal supercategory. A left tensor ideal of C is an ideal I of C satisfying the following condition:

• Given f ∈ I(x, y) and an object z, we have idz ∐ f ∈ I(z ∐ x, z ∐ y). ′ ′ ′ ′ We note that if f ∈ I(x, y) and g ∈ HomC(x ,y ) then g ∐ f ∈ I(x ∐ x ,y ∐ y ).  Fix, for the remainder of this section, a monoidal supercategory (C, ∐). We now introduce a number of conditions on C. (C1) We have a bijection N → Ob(C), denoted n 7→ [n], such that [n] ∐ [m] = [n + m]. (C2) We have HomC([n], [m]) = 0 for n =6 m. (C3) The algebra Bn = EndC([n]) is finite dimensional and semisimple. We note that the monoidal operation induces a superalgebra homomorphism

ιn,m : Bn ⊗ Bm → Bn+m 10 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

for all n, m. In fact, giving C, subject to (C1)–(C3), is equivalent to giving the sequence (Bn)n≥0 of semisimple superalgebras together with the ι maps, such that certain conditions hold. C Let Λ( )= n≥0 Λ(Bn). Let K(Bn) be the Grothendieck group discussed in the previous section, and let K(C)= K(B ). We note that K(C) is a free abelian group with basis ` n≥0 n [S ] with λ ∈ Λ. The monoidal structure induces a ring structure on K(C); precisely, if M is λ L a Bm-module and N is a Bn-module then

[M] · [N] = [Bn+m ⊗Bm⊗Bn (M ⊗ N)], where the tensor product is formed via ιm,n. Note that this tensor product is exact in M and N since the B’s are semisimple, and thus is well-defined on the Grothendieck group. Our next assumption is an analog of the Pieri rule:

(C4) There is a partial order ⊂ on Λ(C) such that for any λ ∈ Λ(Bn) we have

[B1] · [Sλ]= cλ,µ[Sµ] λ⊂µ, µX∈Λ(Bn+1) for positive integers cλ,µ. λ For λ ∈ Λn, we let J ⊂ Bn be the Sλ-isotypic piece of Bn, as in the previous section. If I is a left tensor ideal of C then In = I([n], [n]) is a 2-sided ideal of Bn; moreover, I is determined by the In’s due to (C2). λ Theorem 3.5. Suppose conditions (C1)–(C4) above hold. Let λ ∈ Λ(Bn) and let I be the C λ λ µ left tensor ideal of generated by J . Then Im = λ⊂µ, µ∈Λ(Bm) J for any m.

Given a 2-sided ideal J of Bn, we let Σ(J) beL the 2-sided ideal of Bn+1 generated by ι1,n(1 ⊗ J). The following is the key observation: λ µ Lemma 3.6. Given λ ∈ Λn, we have Σ(J )= λ⊂µ, µ∈Λ(Bn+1) J . λ λ Proof. Let J be the left ideal of Bn+1 generatedL by ι1,n(1 ⊗ J ). Then Σ(J ) is the 2-sided ideal generated by J. The natural map λ Bn+1 ⊗B1⊗Bn (B1 ⊗ J ) → J is an isomorphism: indeed, it is surjective by definition, and it is injective since ι1,n is flat. λ λ We thus have [J] = [B1][J ]. Since [J ]= n[Sλ] for some n> 0, we see that [J]= n[B1][Sλ]. By (C4), it follows that Sµ is a constituent of J if and only if λ ⊂ µ and µ ∈ Λ(Bn+1). Thus µ the 2-sided ideal generated by J is µ∈Λ(Bn+1) J (Proposition 3.2), which completes the proof.  L For m ≥ 0, we let Σm be the m-fold iterate of Σ. m λ µ Lemma 3.7. Given λ ∈ Λn, we have Σ (J )= λ⊂µ, µ∈Λ(Bn+m) J . Proof. This follows from applying the previous lemmaL iteratively.  ′ m−n λ Proof of Theorem 3.5. Let λ ∈ Λ(Bn) be given. Define Im = Σ (J ), which we take to ′ C ′ be 0 for m < n. We claim that I is a left tensor ideal of . Since Im is a 2-sided ideal of Bm for all m, it follows that I′ is closed under arbitrary compositions in C, that is, I′ is an ideal of C. Now, if f ∈ Bm then id[k] ∐ f = ιk,n(1 ⊗ f). It follows that if f belongs to a 2-sided k ′ ′ ideal J of Bm then id[k] ∐ f belongs to Σ (J). In particular, if f ∈ Im then id[k] ∐ f ∈ Im+k. Thus the claim holds. ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 11

Now, let Iλ be the left tensor ideal generated by J λ. Since I′ is a left tensor ideal containing λ λ ′ λ λ λ J , we have I ⊂ I . Since I is a left tensor ideal, we have Σ(Im) ⊂ Im+1 for all m, and so it follows (by a simple inductive argument) that I′ ⊂ Iλ. We thus have Iλ = I′, and so λ m−n λ  Im = Σ (J ). Thus the theorem follows from Lemma 3.7. 3.4. Opposite categories. Suppose that B is a superalgebra. We define the opposite superalgebra, denoted Bop, as follows. The super vector space underlying Bop is just that underlying B. The multiplication • on Bop is defined on homogeneous elements x and y by x • y =(−1)|x||y|yx. Now suppose that C is a supercategory. We define the opposite supercategory, denoted op op C , as follows. The objects of C are the same as the objects as C. We put HomCop (x, y)= HomC(y, x). Given homogeneous morphisms f : x → y and g : y → z, their composition in op |f||g| op C is defined to be (−1) g ◦ f. We note that EndCop (x) = EndC(x) .

3.5. From tensor ideals to algebra ideals. Let C be a small supercategory. A C-module is a superfunctor C → SVec. (A superfunctor is just a functor that is SVec◦-enriched, i.e., it preserves homogeneity and degree of morphisms.) We let ModC be the supercategory of ◦ C-modules. The category ModC is abelian. Suppose now that C has a monoidal structure ∐. Then ModC admits a natural monoidal ◦ structure ⊗, namely Day convolution. Thus ModC is a monoidal category, and so we can speak of algebras, modules over algebras, ideals in algebras, and so on. We note that to give ◦ an algebra in ModC amounts to giving a C-module B with even maps B(x)⊗B(y) → B(x∐y) for all objects x and y, satisfying various conditions. The supercategory Cop ⊠ C is naturally monoidal. (Here ⊠ denotes the product of en- riched categories, see [SS4, §2.5].) This category admits a module R defined by R(x, y) = HomC(x, y). Furthermore, the module R admits an algebra structure: the map R(x, y) ⊗ R(x′,y′) → R(x ∐ x′,y ∐ y′) is simply given by f ⊗ g 7→ f ∐ g. Proposition 3.8. There is a natural bijective correspondence {left ideals of R} ←→{left tensor ideals of C}. Proof. Suppose that I is a left ideal of R. Then I is in particular a C-submodule of R, i.e., a sub superfunctor of R. This exactly means that I is an ideal of the category C in the sense of Definition 3.3. Since I is a left ideal of R, we see that if f ∈ R(x, y) and g ∈ I(x′,y′) then f ∐ g belongs to I(x ∐ x′,y ∐ y′). This shows that I is a left tensor ideal of C. The above reasoning is reversible, and so the proposition follows. 

3.6. The Hecke–Clifford algebras. The Clifford algebra Cln is the superalgebra generated 2 by odd-degree elements α1,...,αn subject to the relations αi = 1 and αiαj = −αj αi for i =6 j. The symmetric group Sn acts on Cln by σ · αi = ασ(i) and we define the Hecke– Clifford algebra Hn to be the semi-direct product C[Sn]⋉Cln. This is again a superalgebra, where we define |σ| = 0 for σ ∈ Sn. The algebra Hn is semisimple [CW1, (3.25)]. The isomorphism classes of simple Hn- modules are parametrized by strict partitions of n [CW1, Proposition 3.41]. We thus identify Λn = Λ(Hn) with the set of such partitions. Given λ ∈ Λn, we have an even isomorphism ∼ Sλ = Sλ[1] if and only if δ(λ) = 1 [CW1, Corollary 3.44]. 12 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

† We define a linear map (−) : Hn → Hn, called transpose, by the formula 2 k † k −1 (2) k −1 (αi1 ··· αik σ) = ζ σ αik ··· αi1 =(−1) ζ σ αik ··· αi1 ,

where here σ ∈ Sn. One readily verifies that this is well-defined. Hop H Proposition 3.9. The transpose map defines an isomorphism n → n of superalgebras. † |x||y| † † Proof. We must show that (xy) = (−1) y x for homogeneous elements x, y ∈ Hn. If x † |x|2 −1 is a basis vector (i.e., of the form αi1 ··· αik σ) then we have x = ζ x . Suppose that x and y are basis vectors of degrees n and m. Then xy is a basis vector of degree n + m. We have 2 (xy)† = ζ(n+m) (xy)−1 2 2 (−1)nmy†x† = ζ2nm+n +m y−1x−1. These are equal. We have thus verified the formula when x and y are basis vectors, from which the general case follows.  H ∗ Hop Suppose that M is an n-module. Then the dual vector space M is naturally an n - ∗ module. Composing with the transpose isomorphism, we can regard M as an Hn-module; we denote this module by M †. 3.7. The Hecke–Clifford category. We now assemble the Hecke–Clifford algebras into a supercategory C. The objects of C are formal symbols [n] with n ∈ N. We put EndC([n]) = Hn and HomC([n], [m]) = 0 for n =6 m. Composition is defined in the obvious manner. We now define a monoidal structure ∐ on C. On objects, we put [m] ∐ [n] = [m + n]. For m, n ≥ 0, we define ιm,n : Hm ⊗ Hn → Hm+n by αi ⊗ 1 7→ αi, 1 ⊗ αj 7→ αm+j, and using the usual embedding Sm × Sn ⊂ Sm+n. One readily verifies that this is a homomorphism of superalgebras. The monoidal operation ∐ is defined on morphisms using ιm,n. We now define a symmetry β on the monoidal structure ∐. Let τm,n ∈ Sm+n be the permutation defined by i + n if 1 ≤ i ≤ m τm,n(i)= . (i − m if m +1 ≤ i ≤ m + n

One readily verifies that for homogeneous elements x ∈ Hm and y ∈ Hn we have −1 mn τm,nιm,n(x ⊗ y)τm,n =(−1) ιn,m(y ⊗ x).

We define βm,n :[m] ∐ [n] → [n] ∐ [m] to be the element τm,n ∈ EndC([m + n]). Put Λ = Λ(C), which we identify with the set of all strict partitions. The group K(C) is the free abelian group on the classes [Sλ] with λ ∈ Λ. We let ⊂ be the partial order on Λ given by containment of Young diagrams; explicitly, λ ⊆ µ means λi ≤ µi for all i. We have the following version of the Pieri rule (recall that δ(λ) is 0 or 1 depending on if ℓ(λ) is even or odd, respectively): Proposition 3.10. Let λ be a strict partition of n. We then have δ(λ) [H1][Sλ]= 2[Sµ]+ 2 [Sµ] µ⊃λ µ⊃λ ℓ(µX)=ℓ(λ) ℓ(µ)=Xℓ(λ)+1 ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 13

in K(C), where µ varies over strict partitions of n +1.

Proof. Let Qλ be the Schur Q-functions indexed by strict partitions λ, see [CW1, §A.3]. They are linearly independent and their Z-span is a subring Γ. There is a characteristic map ch: K(C) ⊗ Q → Γ ⊗ Q, which is an isomorphism of graded algebras (see [CW1, §3.3]). (δ(λ)−ℓ(λ))/2 We have ch([Sλ]) = 2 Qλ and hence ch([H1]) = Q1 [CW1, §3.3.6]. Since ch is an isomorphism, the result follows from the following identity [Ma, III.8.7 and III.8.15]

Q1Qλ = 2Qµ + Qµ.  µ⊃λ µ⊃λ ℓ(µX)=ℓ(λ) ℓ(µ)=Xℓ(λ)+1 Proposition 3.11. The category C satisfies (C1)–(C4). Proof. Conditions (C1) and (C2) are essentially by definition, and (C3) was discussed in the previous section. Condition (C4) follows from Proposition 3.10.  We define the transpose functor (−)† : Cop → C to be the identity on objects and the † transpose operation (−) on each algebra Hn. Proposition 3.12. The transpose functor is naturally an equivalence of symmetric monoidal categories. Proof. It is clear that the transpose functor is indeed a functor and an equivalence. To define a monoidal structure on the transpose functor, we take the isomorphism ([n] ∐ [m])† → [n]† ∐ [m]† to be the identity map; note that both objects are [n + m]. For this to define a monoidal structure, we need † † † ιm,n(f ⊗ g) = ιm,n(f ⊗ g ) H H H H Hop for f ∈ m and g ∈ n. Both sides define algebra m ⊗ n → n+m, so it suffices to check the equality on algebra generators, where it is clear. Finally, for the op † op Cop symmetry condition, we need (βm,n) = βm,n, where β denotes the symmetry in . By op Cop C definition, the morphism βm,n in is equal to the morphism βn,m in . Thus the condition †  we require is τn,m = τm,n, which is true. Let M be a C-module. Then x 7→ M(x)∗ is a Cop-module. Composing with the transpose equivalence, we get a C-module that we denote by M †. If we regard M as a sequence H † † † (Mn)n≥0, where Mn is an n-module, then M is just the sequence (Mn)n≥0, where Mn is as † op in the previous section. The construction (−) defines an equivalence ModC → ModC that is compatible with tensor products. It thus induces a ring automorphism of K(C) that we denote by (−)†. Proposition 3.13. We have (−)† = id on K(C).

Proof. Let Vn = Cln, which is naturally a simple left Hn-module [CW1, §3.3.5]. Under the characteristic map (see the proof of Proposition 3.10), the class [Vn] in K(C) corresponds to the function Q(n) [CW1, Lemma 3.40]. Since the Q(n) generate Γ ⊗ Q as a Q-algebra, it follows that the [Vn] generate K(C) ⊗ Q as a Q-algebra. It thus suffices to show that † † H [Vn] = [Vn]. For this, it is enough to show that Vn and Vn are isomorphic as n-modules. Let t: Cln → C be the map satisfying t(1) = 1 and t(m) = 0 for any monomial m =6 1 in the αi’s. This map satisfies t(xy) = t(yx) for x, y ∈ Cln (there is no sign). For x ∈ Vn, let ∗ † ∗ λx ∈ Vn be defined by λx(y)= t(xy ). We thus have a linear map λ: Vn → Vn via x 7→ λx, 14 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

∗ which is easily seen to be an isomorphism of super vector spaces. Recall that Vn is a left † Hn-module via (aλ)(y)= λ(a y). For a ∈ Cln, we have

† † † † † λax(y)= t(axy )= t(xy a)= t(x(a y) )= λx(a y)=(aλx)(y),

and so λax = aλx. We thus see that λ is a homomorphism of left Cln-modules. Since t is Sn-invariant, it follows that λ is also Sn-equivariant. We thus see that λ is an isomorphism of left Hn-modules, which completes the proof.  † ∼ H Corollary 3.14. We have Sλ = Sλ as n-modules for all λ ∈ Λn. Let R be the canonical Cop ⊠C algebra defined in §3.5. Let R′ be the C⊠C algebra obtained by applying transpose to the first argument in R. Let U be the quotient of Cl1 ⊗ Cl1 by the ′ ∼ 2-sided ideal generated by α1 ⊗ α1 − ζ. One easily sees that R ([1], [1]) = U.

Proposition 3.15. The identification U = R([1], [1]) induces an isomorphism Sym(U) → R′ of C ⊠ C algebras.

n ′ Proof. We first construct an isomorphism ϕn : Sym (U) → R ([n], [n]) of Hn ⊗ Hn-modules ′ for each n. For simplicity of notation, we denote Hn ⊗ Hn by Cn. We note that R ([n], [n]) can be identified with the Cn-module Hn given on generators by

(a ⊗ b).x =(−1)|a|(|x|+|b|)bxa†.

n In particular, we identify U with the C1-module H1. Thus, the Cn-module Sym (U) is given ⊗n ⊗n by the Sn-coinvariants of Cn ⊗ ⊗n H where C → Cn is the natural map that identifies C1 1 1 ⊗n H⊗n C1 with the subalgebra Cln ⊗ Cln of Cn. Under the latter identification, 1 is isomorphic H n H⊗n ∼ to the submodule Cln of the Cn-module n. Thus we have Cn ⊗ ⊗ = Cn ⊗Cln⊗Cln Cln C1 1 as Cn-modules. As a complex vector space, Cn ⊗Cln⊗Cln Cln has a basis of elements of the ǫ1 ǫn form σ ⊗ τ ⊗ α1 ··· αn where σ, τ ∈ Sn, ǫi ∈ {0, 1}. The action of Sn on the nth tensor power Cn ⊗Cln⊗Cln Cln is given by

ǫ1 ǫn −1 −1 ǫ1 ǫn g g.(σ ⊗ τ ⊗ α1 ··· αn )= σg ⊗ τg ⊗ (α1 ··· αn ) ,

n where g ∈ Sn. Thus its Sn-coinvariants, Sym (U), is the free complex vector space on ǫ1 ǫn n H symbols of the form 1⊗τ ⊗α1 ··· αn . Let ϕn : Sym (U) → n be the C-linear isomorphism given by

ǫ1 ǫn ǫ1 ǫn 1 ⊗ τ ⊗ α1 ··· αn 7→ τα1 ··· αn .

ǫ1 ǫn We show that ϕn is Cn-linear. Denote α1 ··· αn by c, and let c1σ1,c2σ2 be two basis elements of Hn. We have

(c1σ1 ⊗ c2σ2)(1 ⊗ τ ⊗ c)=(c1σ1 ⊗ c2σ2τ ⊗ c) −1 −1 σ1 (σ2τ) =(σ1 ⊗ σ2τ)(c1 ⊗ c2 ⊗ c) 2 −1 −1 |c1| |c1|(|c|+|c2|) (σ2τ) −1 σ1 = ζ (−1) σ1 ⊗ σ2τ ⊗ c2 c(c1 ) 2 −1 |c1| |c1|(|c|+|c2|) −1 σ1(σ2τ) σ1 −1 = ζ (−1) 1 ⊗ σ2τσ1 ⊗ c2 c c1 , ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 15

2 −1 |c1| |c1|(|c|+|c2|) σ2τ −1 σ2τσ1 −1 which under ϕn maps to ζ (−1) c2c (c1 ) σ2τσ1 . On the other hand in Hn, we have τ (c1σ1 ⊗ c2σ2)ϕ(1 ⊗ τ ⊗ c)=(c1σ1 ⊗ c2σ2)c τ 2 |c1| |c1|(|c|+|c2|) τ −1 −1 = ζ (−1) c2σ2c τσ1 c1 2 −1 |c1| |c1|(|c|+|c2|) σ2τ −1 σ2τσ1 −1 = ζ (−1) c2c (c1 ) σ2τσ1 .

Thus shows that ϕn is Cn-linear. Now, both Sym(U) and R′ are concentrated on the diagonal of C ⊠ C, that is, they are only non-zero on objects of the form ([n], [n]). Thus the ϕn’s define an isomorphism ϕ: Sym(U) → R′ of C ⊠ C modules. To complete the proof, we simply observe that ϕ is an algebra homomorphism. Indeed, suppose that x ∈ Symn(U) and y ∈ Symm(U). Write ǫ1 ǫn δ1 δm x =1 ⊗ σ ⊗ α1 ··· αn and y =1 ⊗ τ ⊗ α1 ··· αm . Then one finds ǫ1 ǫn δ1 δm xy =1 ⊗ στ ⊗ α1 ··· αn αn+1 ··· αn+m, (where here τ is regarded as a permutation of {n +1,...,n + m}) and so

ǫ1 ǫn δ1 δm ϕn+m(xy)= στα1 ··· αn αn+1 ··· αn+m,

which is exactly ϕn(x)ϕm(y)= ιn,m(ϕn(x),ϕm(y)).  Given two C-modules M and N, we let M ⊠ N denote the C ⊠ C module given by (x, y) 7→ M(x) ⊗ N(y). If M is an Hn-module, we regard M as a C-module that is 0 outside of degree n. We now come to the main result of this section: −δ(λ) Theorem 3.16. The C ⊠ C algebra Sym(U) decomposes as 2 (Sλ ⊠ Sλ), where the sum is over all strict partitions λ. Moreover, if Iλ denotes the ideal of Sym(U) generated by the λ summand then Iλ contains the µ summand if and only ifLλ ⊂ µ.

Proof. The algebra Hn is naturally an (Hn, Hn)-bimodule, and as such it decomposes as λ H H λ∈Λn J (Proposition 3.1). We can regard any ( n, n)-bimodule as a left module over Hop H Cop ⊠C n ⊗ n, and thus as a module concentrated in degree (n, n). With this convention, Lwe have the decomposition of Cop ⊠ C modules λ R = Hn = J . n≥0 M Mλ∈Λ The first identification above is essentially the definition of R, while the second follows from the decomposition of Hn. It follows from Theorem 3.5 and Proposition 3.8 that the ideal of R generated by J λ contains exactly those J µ with λ ⊂ µ. λ Hop H As stated above, J can be regarded as a left module over n ⊗ n. As such, it is ∗ isomorphic to the submodule of End(Sλ)= Sλ ⊗Sλ consisting of endomorphisms that super- commute with EndHn (Sλ); this follows from [CW1, (3.4)]. If δ(λ) = 0 then EndHn (Sλ)= C, λ ∼ ∗ and we find J = Sλ ⊗ Sλ. Suppose now that δ(λ) = 1. Then there exists an isomorphism Sλ → Sλ[1]. From this, we find that there is an odd isomorphism α: Sλ → Sλ that squares to 1 (note that any odd isomorphism squares to a scalar, by the super version of Schur’s lemma [CW1, Lemma 3.4], and can thus be normalized to square to 1). The dual map ∗ ∗ ∗ λ ∗ α : Sλ → Sλ squares to −1. We see that J is isomorphic to the 1-eigenspace of α ⊗ α ∗ ∗ ∗ ∗ on Sλ ⊗ Sλ. This coincides with the ζ-eigenspace of (ζα ) ⊗ α. As ζα endows Sλ with an λ −1 ∗ isomeric structure, we see that J is isomorphic to the half tensor product 2 (Sλ ⊗ Sλ). λ ∼ −δ(λ) ∗ Thus, in all cases, we see that J = 2 (Sλ ⊗ Sλ). 16 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

λ Via transpose on the first factor, we can regard J as an Hn ⊗ Hn module. As such, −δ(λ) † we see that it is isomorphic to 2 (Sλ ⊗ Sλ), which (by Corollary 3.14) is isomorphic to −δ(λ) 2 (Sλ ⊗ Sλ). Combining this observation with the decomposition in the first paragraph, we find ′ −δ(λ) R = 2 (Sλ ⊠ Sλ). Mλ∈Λ Furthermore, we see that the ideal generated by the λ summand contains the µ summand if and only if λ ⊂ µ. The theorem now follows from the identification R′ = Sym(U) (Proposi- tion 3.15).  3.8. Application to the isomeric algebra A. Recall that V = C∞|∞ is our fixed infinite dimensional isomeric vector space. Maintaining the notation from §2.8, we let U =2−1(V ⊗ V), which is a polynomial representation of q × q, and we let A = Sym(U), a bivariate isomeric algebra. As noted in §2.8, we have a Cauchy decomposition −δ(λ) (3.17) A = 2 (Tλ ⊗ Tλ), Mλ∈Λ where Λ is the poset of strict partitions, as above, and δ(λ) is as in §2.5. The follow theorem, which classifies the equivariant ideals of A, is one of the main results of this paper: Theorem 3.18. Suppose λ is a strict partition, and let Iλ be the ideal of A generated by −δ(λ) λ −δ(µ) 2 (Tλ⊗Tλ). Then we have I = λ⊆µ 2 (Tµ⊗Tµ) where µ varies over strict partitions. Proof. One formulation of Schur–SergeevL duality (see [SS4, §8.2]) asserts that there is an pol equivalence of symmetric monoidal supercategories ModC → Rep (q) satisfying Sλ 7→ Tλ. Here ModC is endowed with the tensor product induced by the monoidal structure on C (Day convolution). It follows that we have an equivalence of symmetric monoidal supercategories pol Φ: ModC⊠C → Rep (q × q) −1 satisfying Φ(Sλ ⊠ Sµ)= Tλ ⊗ Tµ. Since U =2 (S(1) ⊠ S(1)), it follows that U = Φ(U), and A = Φ(Sym(U)). The result now follows from Theorem 3.16.  Theorem 3.18 has an important consequence: Corollary 3.19. If I ⊂ A is a nonzero ideal, then A/I is bounded. Proof. Since I =6 0, there exists a strict partition λ such that Iλ ⊆ I. Suppose that µ is a strict partition such that ℓ(µ) ≥ λ1. Then µi ≥ ℓ(µ) − i +1 ≥ λ1 − i +1 ≥ λi and hence λ ⊆ µ. So Theorem 3.18 implies that ℓ(A/I) <λ1.  Remark 3.20. Let X = Spec(A(Cn|n, Cm|m)), which we regard as a supervariety (i.e., a functor from superalgebras to sets). We can identify points of X with linear maps Cn|n → Cm|m commuting with the isomeric structures. The image of any such map is isomorphic to r|r C for some r, which we refer to as the rank. Let Ir ⊂ A be the sum of the λ summands in the Cauchy decomposition with ℓ(λ) > r. Then Ir is an equivariant ideal of A. It r|r follows from properties of the Tλ (namely, that Tλ(C ) = 0 if and only if ℓ(λ) > r) that n|n m|m V (Ir(C , C )) is the closed subvariety of X consisting of maps of rank ≤ r. Thus Ir is an isomeric analog of the classical determinantal ideal. It follows from Theorem 3.18 that Ir is generated by the λ summand of the Cauchy decomposition, where λ is the “staircase” partition (r +1, r, . . . , 2, 1).  ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 17

4. Local structure of A-modules

4.1. Statement of results. Let m be the ideal of A generated by xi,j − δi,j and yi,j for all i and j. In this section, we study the local structure of A-modules at m. Our main result is: Theorem 4.1. Let M be an A-module. Then there is a canonical and functorial isomorphism Mm → M/mM ⊗C Am of |Am|-modules. In particular, Mm is a free |Am|-module. The proof will take the rest of the section. The arguments are similar to those used in [SS2, §4], [NSS1, §3.2], and [NSS2, §5]. Theorem 4.1 is a key result used to study generic A-modules in the following section. 4.2. The algebras h and k. Let h be the subalgebra of q × q consisting of all pairs of the form (τg,g), where τ is the Chevalley automorphism of q × q. Note that h is isomorphic to q (via the first projection). Let b (resp. n) be the subalgebra of q consisting of matrices of the form a b . −b a   with a and b upper triangular (resp. strictly upper triangular), and let k = b × n, regarded as a subalgebra of q × q. Proposition 4.2. We have q × q = h ⊕ k. Proof. To simplify notation, put a b {a, b} = . −b a   Let a1, a2, b1, and b2 be given matrices. We must show that there are unique matrices c1, c2, d1, d2, e1, e2, with d1,d2 upper triangular and e1, e2 strictly upper triangular such that −1 ({a1, a2}, {b1, b2})=({c1,c2}, τ {c1,c2})+({d1,d2}, {e1, e2}). The above equation is equivalent to t a1 = c1 + d1 b1 = −c1 + e1 t a2 = c2 + d2 b2 = ζc2 + e2. Apply transpose to the right two equations and multiply the bottom right equation by ζ; then add the left column to the right. We obtain t t a1 = c1 + d1 a1 + b1 = d1 + e1 t t a2 = c2 + d2 a2 + ζb2 = d2 + ζe2 The two right equations have unique solutions, as every matrix can be written uniquely as the sum of an upper triangular matrix and a strictly lower triangular matrix. The two left equations then obvious admit unique solutions as well.  Proposition 4.3. The ideal m is h-stable.

Proof. Let Ei,j be the elementary matrix whose (i, j) entry is 1 and all other entries are 0; thus Ei,j maps the jth basis vector to the ith basis vector. Let E 0 0 E X = i,j ,Y = i,j . i,j 0 E i,j −E 0  i,j  i,j  18 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

The Xi,j and Yi,j elements form a basis for q. We have

Xi,j(ek)= δj,kei Yi,j(ek)= −δj,kfi

Xi,j(fk)= δj,kfi Yi,j(fk)= δj,kei. ′ −1 ′ −1 The elements Xi,j = (Xi,j, τ Xi,j) and Yi,j = (Yi,j, τ Yi,j) form a basis for h. Note that −1 −1 τ Xi,j = −Xj,i and τ Yi,j = ζYj,i. A simple computation gives ′ ′ Xi,j(vk,ℓ)= δj,kvi,ℓ − δi,ℓvk,j Yi,j(vk,ℓ)= −ζ · (δj,kwi,ℓ + δi,ℓwk,j) ′ ′ Xi,j(wk,ℓ)= δj,kwi,ℓ − δi,ℓwk,j Yi,j(wk,ℓ)= −ζ · (δj,kvi,ℓ − δi,ℓvk,j)

Recall that the xi,j transform exactly like the vi,j, and the yi,j transform exactly like the wi,j. Note that

δj,kxi,ℓ − δi,ℓxk,j = δj,k(xi,ℓ − δi,ℓ) − δi,ℓ(xk,j − δk,j). ′ ′ We thus see that Xi,j and Yi,j map each generator of m to a linear combination of generators, which completes the proof.  Remark 4.4. The space Spec(A) parametrizes bilinear forms h, i: V ⊗ V → C satisfying hα(v),α(w)i =(−1)|v|ζhv,wi. The ideal m corresponds to a standard form, and the above proposition simply says that the adjoint of an element X of q with respect to this form is given by −τ −1X.  Lemma 4.5. If a is a non-zero ideal of A, then a + m = A.

n|n Proof. Suppose a is a non-zero ideal of A. Let Vn and Wn be copies of C and let An = −1 Sym(2 (Vn ⊗ Wn)), regarded as a subring of |A|. Then A is the union of the An, and so ′ ′ for n ≫ 0, a = a ∩ An is a non-zero q(n) × q(n)-stable ideal of An, and m = m ∩ An is a maximal ideal of An. By Remark 4.4, Spec(An) is a space of certain bilinear forms on ′ Vn, and m ∈ Spec(An) has maximal rank. Since every ideal contains non-zero even degree ′ elements, V (a ) is a proper closed q(n) × q(n)-stable subset of Spec(An). Hence, it cannot contain any form of maximal rank (as the orbit of any such form is dense under the action of ′ ′ gl(n) × gl(n)=(q(n) × q(n))0), and so a 6⊂ m . It follows that a 6⊂ m, and so a + m = A. 

4.3. The group K. Let B be the supergroup consisting of all infinite matrices of the form a b −b a   with a and b upper triangular, a even, b odd, and a invertible; we do not require a or b to fix almost all basis vectors. This is naturally a group object in the category of super schemes. −1 The coordinate ring C[B] is the superalgebra C[ai,j, bi,j, ai,i ]j≥i where the a’s are even and the b’s are odd. If we multiply two matrices in B then each entry of the result involves only finitely many entries of the original two matrices; the formulas for the entries of the result are used to define the comultiplication map C[B] → C[B]⊗C[B]. We let U be the subgroup of B where ai,i = 1 and bi,i = 0 for all i, and we let K = B × U. We use coordinates ai,j, bi,j, ′ ′ ai,j, bi,j on K, where the first pair are coordinates on B the second on U. The Lie algebra of B is essentially the algebra b defined in the previous section, except its elements can have infinitely many non-zero entries. Similarly, the Lie algebra of U is essentially n, and that of K is essentially k. ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 19

If V is a polynomial representation of q then its restriction to b naturally extends to a representation of B (i.e., it admits a comodule structure over C[B]). Similarly, if V is a poly- nomial representation of q × q then its restriction to k naturally extends to a representation of K.

4.4. The map ϕ. Let M be an A-module. We then have the comultiplication map M → M ⊗ C[K] as discussed above. Composing with the quotient map M → M/mM, we obtain a linear map

ϕM : M → M/mM ⊗ C[K].

In fact, we can define ϕM for any K-equivariant |A|-module (where “K-equivariant” means the action of K comes from a comodule structure). We now study this map, beginning with the case M = A: Proposition 4.6. We have the following:

(a) The map ϕA : A → C[K] is the C-algebra homomorphism given by

′ ′ ′ ′ ϕ(xi,j)= ak,iak,j + ζbk,ibk,j, ϕ(yi,j)= ak,ibk,j − ζbk,iak,j kX≤i,j kX≤i,j (b) The extension of m along the map ϕA is the maximal ideal of C[K] corresponding to the identity element of K. (c) The map ϕA induces an isomorphism of localizations Am → C[K]m.

′ ′ Proof. (a) Let g be an element of K with coordinates ai,j, bi,j, ai,j, bi,j. We have

′ ′ g(ei ⊗ ej)= ak,iek − bk,ifk ⊗ aℓ,jeℓ − bℓ,jfℓ ! ! Xk≤i Xℓ≤j ′ ′ g(ei ⊗ fj)= ak,iek − bk,ifk ⊗ bℓ,jeℓ + aℓ,jfℓ ! ! Xk≤i Xℓ≤j ′ ′ g(fi ⊗ ej)= bk,iek + ak,ifk ⊗ aℓ,jeℓ + bℓ,jfℓ ! ! Xk≤i Xℓ≤j ′ ′ g(fi ⊗ fj)= bk,iek + ak,ifk ⊗ −bℓ,jeℓ + aℓ,jfℓ ! ! Xk≤i Xℓ≤j

We thus find ′ ′ ′ ′ gvi,j = (ak,iaℓ,j + ζbk,ibℓ,j)vk,ℓ − (ak,ibℓ,j + ζbk,iaℓ,j)wk,ℓ k≤Xi,ℓ≤j ′ ′ ′ ′ gwi,j = (ak,ibℓ,j − ζbk,iaℓ,j)vk,ℓ +(ak,iaℓ,j − ζbk,ibℓ,j)wk,ℓ k≤Xi,ℓ≤j from which the stated formulas for ϕ(xi,j) and ϕ(yi,j) follow (change vi,j to xi,j and wi,j to yi,j, and reduce modulo m, which takes xi,j to δi,j and yi,j to 0). Since K acts on A by algebra homomorphisms, it follows that ϕ is an algebra homomorphism. 20 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

′ ′ e (b) It suffices to show that the elements ai,j − δi,j, bi,j, ai,j and bi,j are in the extension m of m. We do this lexicographically on the pair (i, j) (with j being more significant). The base case (1, 1) follows from:

ϕ(x1,1 − 1) = a1,1 − 1

ϕ(y1,1)= −ζb1,1. Suppose the result hold for all pairs smaller than (i, i). Then we have the following: e ϕ(xi,i − 1) = ai,i − 1 (mod m ) e ϕ(yi,i)= −ζbi,i (mod m ), e from which it follows that ai,i − 1, bi,i ∈ m . Next suppose i < j and assume, by induction, that the result holds for all pairs smaller than (i, j). Then we have the following: ′ e ϕ(xi,j)= ai,j (mod m ) ′ e ϕ(yi,j)= bi,j (mod m ) e ϕ(xj,i)= ai,j (mod m ) e ϕ(yj,i)= −ζbi,j (mod m ). ′ ′ e It follows that ai,j, bi,j, ai,j, bi,j ∈ m , which proves (b). (c) We now define an algebra morphism ψ : C[K] → Am such that the kernel of the e composition C[K] → Am → Am/mAm is the ideal m . We define ψ lexicographically on the generators as follows. In the base case, we set

ψ(a1,1)= x1,1

ψ(b1,1)= ζy1,1. Suppose i ≤ j and assume, by induction, that ψ has been defined on all generators indexed smaller than (i, j). We set

′ ′ ψ(ai,j)= xj,i − ψ(ak,jak,i + ζbk,jbk,i) Xk

′ −1 ′ ′ ψ(ai,j) ψ(ai,i) ζψ(bi,i) xi,j − k

that ϕM is a morphism of A-modules. Our goal is to prove that ϕM induces an isomorphism after localizing at m, which will establish the theorem. We require some lemmas first.

Lemma 4.7. Let M be a K-equivariant |A|-module. Then ϕM induces an isomorphism modulo m. Proof. See [SS2, Lemma 4.4]. 

Lemma 4.8. Let M be an A-module. Then the localization of ϕM at m is surjective. Proof. See [SS2, Lemma 4.5]. 

Lemma 4.9. Let M be an A-module. Then the kernel of ϕM is q × q-stable, and thus an A-submodule of M. Proof. This is proved exactly like [SS2, Lemma 4.8], using an obvious generalization of [SS2, Lemma 4.7] to superalgebras. 

By an “polynomially h-equivariant Am-module,” we mean an Am-module N equipped with a compatible action of h such that for every x ∈ N there is a unit u ∈ Am such that ux generates a polynomial h-representation (recall that h ∼= q). The localization of any A- module at m is a polynomially h-equivariant Am-module. Any submodule or quotient module of a polynomially h-equivariant Am-module (in the category of equivariant modules) is again polynomially equivariant. Lemma 4.10. Suppose that 0 → R → M → N → 0

is an exact sequence of polynomially h-equivariant Am-modules such that M is equivariantly finitely generated and N is free as an |Am|-module. Then R is also equivariantly finitely generated. Proof. The argument in [NSS2, Lemma 5.8] applies here. 

Lemma 4.11. Let M be an A-module such that M = mM. Then Mm =0. Proof. In what follows, q is identified with the subalgebra h ⊂ q × q. It suffices to prove the lemma when M is finitely generated, so we assume this in what follows. Let V ⊂ M be a finite length GL-subrepresentation generating M as an A-module. N|N Pick m1,...,mk ∈ V such that the mi generate V (C ) as a q(N)-representation for all N ≫ 0(V is a sum of finitely many irreducible representations, which are generated by their highest weight spaces, so it suffices to pick N large enough so that each of these highest N|N weight spaces are nonzero, and pick the mi to span the highest weight spaces in V (C )). Write mi = i ai,jni,j where ai,j ∈ m and ni,j ∈ M. ′ N|N Let N ≫ 0 be large enough so that the mi and the ni,j belong to M = M(C ) and P ′ N|N ′ N|N ′ ′ the ai,j belong to A = A(C ). Let V = V (C ) and let m be the ideal of A generated ′ ′ by xi,j − δi,j and yi,j for i, j ≤ N. Then M is an A -module and generated (ignoring any ′ ′ ′ ′ ′ Lie superalgebra action) by V . We have mi ∈ m M for all i, and so gmi ∈ m M for any g ∈ q(N), since m′ is q(N)-stable. Thus V ′ ⊂ m′M ′ and so M ′ = m′M ′. Thus, by the usual ′ version of Nakayama’s lemma [Lam, (4.22)], we have Mm′ = 0. Therefore, for each 1 ≤ i ≤ k ′ ′ there exists homogeneous degree 0 elements si ∈ A \ m ⊂ A \ m such that simi = 0, which implies that Mm = 0.  We now reach the main result: 22 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

Proposition 4.12. Let M be an A-module. Then ϕM induces an isomorphism after local- izing at m. Proof. The argument from [SS2, Proposition 4.11] applies. For the convenience of the reader, we produce it here: The assignment M 7→ ϕM commutes with filtered colimits, and so it suffices to treat the case where M is finitely generated. Let R be the kernel of ϕM , which is an A-submodule of M by Lemma 4.9, and let N = M/mM ⊗ C[K]. By Lemma 4.8, the localization of ϕM at m is a surjection. Since localization is exact, we have an exact sequence of polynomially h-equivariant Am-modules 0 → Rm → Mm → Nm → 0.

From Lemma 4.10, we conclude that Rm is equivariantly finitely generated as an Am-module. Let V ⊂ R be a finite length polynomial q × q-representation generating Rm as an |Am|- module, and let R0 be the A-submodule of R generated by V . Note that R0 is finitely generated as an A-module and (R0)m = Rm. Now, the mod m reduction of the above exact sequence is exact, by the freeness of Nm, and the reduction of Mm → Nm is an isomorphism by Lemma 4.7. We conclude that R/mR = R0/mR0 = 0. Lemma 4.11 thus shows that (R0)m =0 and so Rm = 0, and the proposition is proved. 

5. The generic category 5.1. Algebraic representations of the infinite isomeric algebra. Recall that V = ∞|∞ ∗ C is our fixed isomeric vector space with basis {ei, fi}i≥1. The V is naturally ∗ a representation of q. We define the restricted dual, denoted V∗, to be the subspace of V ∗ ∗ ∗ spanned by the dual basis vectors ei and fi . This is a q-subrepresentation of V , and isomorphic to the direct limit of the spaces (Cn|n)∗. We say that a representation of q is algebraic if it occurs as a subquotient of a (perhaps infinite) direct sum of mixed tensor ⊗n ⊗m alg spaces Tn,m = V ⊗V∗ . We let Rep (q) be the category of algebraic representations, and Repalg,f (q) the subcategory spanned by objects of finite length. This category is studied in detail in [GS], where it is denoted Trep(g); see [GS, Definition 3.2] for the precise definition. (Similar categories for classical Lie algebras were studied in [DPS, PSe, PSt, SS1].) We will require some results from [GS], which we now review. Proposition 5.1. We have the following:

(a) The representation Tn,m has finite length. (b) Every algebraic representation of q is the union of its finite length subobjects. (c) We have Repalg,f (q) = Trep(g). alg (d) For strict partitions λ and µ, there is a simple object Vλ,µ of Rep (q), and every simple object is isomorphic to Vλ,µ or Vλ,µ[1] for some λ and µ. (e) For strict partitions λ and µ, the representation Tλ(V) ⊗ Tµ(V∗) is injective in alg Rep (q); in fact, it is the injective envelope of Vλ,µ. (f) Every finite length object of Repalg(q) has finite injective dimension. Proof. (a) is explained in the paragraph following [GS, Definition 3.2]. (b) follows from (a) and the definition of algebraic representation. We now explain (c). Since Tn,m belongs to Trep(g) (see the paragraph following [GS, Definition 3.2]) and Trep(g) is an abelian subcategory of the category of all q-modules, it follows that Repalg,f (q) ⊂ Trep(g). By [GS, Lemma 3.10] and [GS, Corollary 4.3], every ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 23

simple object of Trep(g) embeds into some Tn,m. By [GS, Proposition 4.10], Tn,m is injective in Trep(g), and so we see that every object of Trep(g) embeds into a sum of Tn,m’s. This gives the reverse inclusion. (d) follows from [GS, Lemma 3.10]. (e) follows from [GS, Proposition 4.11] (this only gives injectivity in Repalg,f (q), but it is easy to see that the representation remains injective in Repalg(q)). (f) follows from [GS, Lemma 5.13].  5.2. Torsion modules. We now define a notion of “torsion” for A-modules. We begin with the following observation: Proposition 5.2. Let M be an A-module. The following conditions are equivalent: (a) For every finitely generated submodule M ′ of M there is a non-zero ideal a of A such that aM =0. (Here “ideal” implies q × q-stable.) (b) We have Mm =0. (c) For every m ∈ M there exists a ∈ A with non-zero image in C[xi,j] = A/(yi,j) such that am =0. Proof. First suppose that (a) holds. Let M ′ ⊂ M be finitely generated with non-zero annihi- ′ ′ ′ lator a. Since a+m = A by Lemma 4.5, we have mM = M , and so Mm = 0 by Lemma 4.11. ′ Since this holds for all finitely generated M ⊂ M, it follows that Mm = 0 and hence (b) holds. Now suppose (b) holds. Given m ∈ M, there exists s ∈ A \ m such that sm = 0. As s has non-zero reduction in C[xi,j], one can take a = s in (c). Thus (c) holds. ′ Finally, suppose (c) holds. Let M be a submodule of M generated by m1,...,mk. Let aimi = 0 with ai as in (c). Let a = a1 ··· ak, which has non-zero image in A/(yi,j) since C[xi,j] is a domain, and annihilates each mi. Following the proof of [NSS1, Prop. 2.2], we n see that there exists n, depending only on the mi, such that a (gmi) = 0 for all g ∈ q × q. Now (a) holds by taking a to be the (non-zero) ideal of A generated by an.  We say that an A-module is torsion if it satisfies the equivalent conditions of the above tors proposition. We write ModA for the category of torsion modules. It is clearly a Serre subcategory of ModA. gen 5.3. Statement of results. We define the generic category ModA to be the Serre quo- tors gen tient ModA / ModA . We write T : ModA → ModA for the localization functor and let S be its right adjoint (the section functor). The goal of this section is to understand the structure of the generic category and the behavior of T and S. We accomplish this by relating the generic category to Repalg(q). Let M be an A-module. Define Φ(M) = M/mM, where m is the maximal ideal of |A| considered in the previous section. Since m is stable under h ⊂ q × q, it follows that Φ(M) is naturally a representation of h ∼= q. It is easily seen to be algebraic: indeed, expressing M as a quotient of A ⊗ V , with V a polynomial representation of q × q, we see that Φ(M) is a quotient of V |h, and thus algebraic. We have thus defined a functor alg Φ: ModA → Rep (q). Since Φ is cocontinuous and its source and target are Grothendieck abelian categories, it admits a right adjoint Ψ. The following is the main theorem of this section: Theorem 5.3. We have the following: 24 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

(a) The functor Φ is exact. tors (b) The kernel of Φ is ModA . (c) The counit ΦΨ → id is an isomorphism. gen alg (d) The functor Φ induces an equivalence ModA → Rep (q). (e) The unit V ⊗ A → Ψ(Φ(V ⊗ A)) is an isomorphism for any V ∈ Reppol(q × q). Appealing to our knowledge of Repalg(q) (Proposition 5.1), we find: Corollary 5.4. We have the following: (a) If M is a finitely generated A-module then T (A) has finite length. gen (b) Every finite length object of ModA has finite injection dimension. gen pol (c) The injectives of ModA are exactly the objects T (V ⊗ A) with V ∈ Rep (q × q). (d) The unit V ⊗ A → S(T (V ⊗ A)) is an isomorphism, for any V ∈ Reppol(q × q). Theorem 5.3 is analogous to [SS2, Theorem 5.1], [NSS1, Theorem 3.1], [NSS2, Theo- rem 6.1]. We follow the proof of [SS2], which is modeled on the proofs in the other two papers but contains some simplifications. 5.4. An equality of dimensions. We will require the following result in our proof of Theorem 5.3: Proposition 5.5. For strict partitions λ,µ,α,β, we have

dim Homq(V)×q(W)(Tλ(V) ⊗ Tµ(W), Tα(V) ⊗ Tβ(W) ⊗ A)

= dim Homq(V)(Tλ(V) ⊗ Tµ(V∗), Tα(V) ⊗ Tβ(V∗)) Proof. Let λ fµ,ν = dim Homq(V)(Tλ(V), Tµ(V) ⊗ Tν(V)). By (3.17), we have

−1 −δ(γ) Sym(2 (V ⊗ W)) = 2 Tγ(V) ⊗ Tγ(W). γ M We now have two cases. If |λ|−|α|= 6 |µ|−|β|, then the left side is 0 (by [GS, Corollary λ 3.7], if fµ,ν =6 0, then |λ| = |µ| + |ν|). The right side is also 0 by [GS, Corollary 5.11]. Otherwise, suppose that r = |λ|−|α| = |µ|−|β|. Then the left side simplifies to

−δ(γ) λ µ 2 fα,γfβ,γ. |Xγ|=r −δ(λ) This is also the right hand side by [GS, Theorem 5.8] (Z(λ,µ) is defined to be 2 (Tλ(V)⊗ ∗ Tλ(V )) in [GS, §4.1], which accounts for the difference in the form of the formula).  5.5. Proof of Theorem 5.3. The proofs of the following lemmas are nearly identical to those in [SS2, §5.2], so we omit most details. For strict partitions λ and µ, let Fλ,µ = Tλ(V) ⊗ Tµ(V) ⊗ A, and let F be the class of A-modules that are direct sums (perhaps infinite) of Fλ,µ’s. Lemma 5.6. Let f : M → N be a morphism of A-modules such that Φ(f)=0. Then the localized morphism fm : Mm → Nm vanishes. Proof. See [SS2, Lemma 5.3].  ON THE GEOMETRY AND REPRESENTATION THEORY OF ISOMERIC MATRICES 25

Lemma 5.7. For any F ∈ F and any strict partitions λ and µ, the map

Φ: HomA(Fλ,µ, F ) → Homq(Φ(Fλ,µ), Φ(F )) is an isomorphism. Proof. See [SS2, Lemma 5.4], and use Proposition 5.5 in place of [SS2, Proposition 3.9].  Lemma 5.8. Let f : M → N be a morphism of A-modules. Suppose that for all strict partitions λ and µ the induced map

f∗ : HomA(Fλ,µ, M) → HomA(Fλ,µ, N) is an isomorphism. Then f is an isomorphism.

Proof. This follows from the fact that the Fλ,µ generate ModA, see [SS2, Lemma 5.6]. 

Lemma 5.9. For F ∈ F, the unit ηF : F → Ψ(Φ(F )) is an isomorphism. Proof. See [SS2, Lemma 5.7].  alg Lemma 5.10. Let I be an injective object of Rep (q). Then the counit ǫI : Φ(Ψ(I)) → I is an isomorphism. Proof. See [SS2, Lemma 5.8].  The proof of Theorem 5.3 now follows in the same way as the proof of [SS2, Theorem 5.1]. 6. Proof of the noetherianity theorem We can now finally prove the noetherianity theorem: Theorem 6.1. The bivariate isomeric algebra A is noetherian. The proof will occupy the remainder of this section. We say that an A-module M satisfies A (FT) if Tori (M, C) is a finite length q × q-module for all i ≥ 0. This implies M is finitely A generated (it suffices to just know that Tor0 (M, C) is finite length) by Nakayama’s lemma. Lemma 6.2. Let M be a finitely generated torsion A-module. Then M satisfies (FT) and M is noetherian. Proof. Now let M be a finitely generated torsion A-module. Then there is a non-zero ideal I ⊂ A which annihilates M. By Corollary 3.19, A/I is bounded, so M is noetherian by A Proposition 2.2. Next, Tori (M, C) is computed by the Koszul complex, and hence is a subquotient of i(U) ⊗ M, which is bounded. Hence it can be computed by specializing to finitely many variables, which implies that it has finite length.  V gen Lemma 6.3. If M is a finite length object of ModA then S(M) satisfies (FT). Proof. The proof is essentially the same as [NSS1, Proposition 4.8]. We recall the details. We in fact show that (RiS)(M) satisfies (FT) for all i ≥ 0. We proceed by induction on the injective dimension of M, which is finite by Corollary 5.4(b). First suppose that M is injective. Then M = T (V ⊗ A) for some finite length V ∈ Reppol(q×q) (Corollary 5.4), and so S(M)= V ⊗A (Corollary 5.4), which obviously satisfies (FT); of course, (RiS)(M)=0 for i> 0 since M is injective. Thus the claim holds. Now suppose that M has positive injective dimension. Choose a short exact sequence 0 → M → I → N → 0 26 ROHITNAGPAL,STEVENVSAM,ANDANDREWSNOWDEN

where I is injective and N has smaller injective dimension than M. Then (RiS)(M) = (Ri−1S)(N) for i ≥ 2, and thus satisfies (FT) by the inductive hypothesis. We have an exact sequence 0 → S(M) → S(I) → S(N) → (R1S)(M) → 0. Since S(N) satisfies (FT) by the inductive hypothesis, it is finitely generated, and so (R1S)(M) is finitely generated. Since (R1S)(M) is also torsion, it satisfies (FT) by Lemma 6.2. It thus follows that S(M) satisfies (FT), as the other three terms in the exact sequence do. 

Proof of Theorem 6.1. The proof is essentially the same as [NSS1, Theorem 4.9]. We recall the details. Let P be a finitely generated projective A-module and let N1 ⊂ N2 ⊂ ··· be an ascending chain of A-submodules. Then T (N•) is an ascending chain in T (P ), and thus stabilizes, as T (P ) has finite length. Discarding finitely many terms, we assume that T (N•) ′ is constant. Let N be the common value of S(T (Ni)). We have the following: • N ′ is finitely generated by Lemma 6.3; ′ • N is a submodule of P = S(T (P )) that contains each Ni; and ′ ′ • N /N1 is torsion, since T (N /N1) = 0. ′ Since N /N1 is finitely generated and torsion, it is noetherian (Lemma 6.2), so the ascending chain N•/N1 in it stabilizes. It follows that N• stabilizes, and so P is noetherian. 

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Department of Mathematics, University of Michigan, Ann Arbor, MI Email address: [email protected] URL: http://www-personal.umich.edu/~rohitna/ Department of Mathematics, University of California, San Diego, CA Email address: [email protected] URL: http://math.ucsd.edu/~ssam/ Department of Mathematics, University of Michigan, Ann Arbor, MI Email address: [email protected] URL: http://www-personal.umich.edu/~asnowden/