THE COEFFICIENT COALGEBRA OF A SYMMETRIZED TENSOR SPACE RANDALL R. HOLMES DAVID P. TURNER Abstract. The coefficient coalgebra of r-fold tensor space and its dual, the Schur algebra, are generalized in such a way that the role of the symmetric group Σr is played by an arbitrary subgroup of Σr. The dimension of the coefficient coalgebra of a symmetrized tensor space is computed and the dual of this coalgebra is shown to be isomorphic to the analog of the Schur algebra. 0. Introduction Let K be the field of complex numbers. The vector space E = Kn is naturally viewed as a (left) module for the group algebra KΓ of the general ⊗r linear group Γ = GLn(K). The r-fold tensor product E is in turn a module for KΓr, where Γr = Γ × · · · × Γ(r factors). Let G be a subgroup of the symmetric group Σr and let χ : G ! K be an irreducible character χ ⊗r of G. The \symmetrized tensor space" associated with χ is E = E tχ, where tχ is the central idempotent of the group algebra KG corresponding to χ (with the action of G on E⊗r being given by place permutation). In this paper, we study the coefficient coalgebra Aχ of the R-module Eχ, where R is the subalgebra of KΓr consisting of those elements fixed r by G under the action of Σr on Γ given by entry permutation. (If V is an R-module with finite K-basis fvj j j 2 Jg, then for each κ 2 R, we have P κvj = i αij(κ)vi for some αij : R ! K. The linear span of the functions αij (i; j 2 J) has a natural coalgebra structure. It is the \coefficient coalgebra" of V , denoted cf(V ).) Section1 sets up notation and presents some standard results suitably generalized to the current situation. In Section2, we obtain generalizations of some classical results (i.e., re- sults for the case G = Σr). In particular, we show that the image S of the representation afforded by the R-module E⊗r equals the set of those endo- morphisms of E⊗r that commute with the action of G (Theorem 2.7). This generalizes Schur's Commutation Theorem [Ma, 2.1.3] in the classical case, where S is the \Schur algebra." We also observe that the algebra isomor- phism S =∼ A∗ (A = cf(E⊗r)) in the classical case [Ma, 2.3.5 and following paragraph] continues to hold with G arbitrary (Theorem 2.8). Date: April 26, 2011. 1991 Mathematics Subject Classification. 20G43, 16T15. 1 2 RANDALL R. HOLMES DAVID P. TURNER In Section3, we study the coefficient coalgebra Aχ and the corresponding ∼ analog Sχ of the Schur algebra and establish an algebra isomorphism Sχ = ∗ Aχ (Theorem 3.4). We exhibit decompositions of A and S in terms of the various Aχ and Sχ, respectively (Theorems 3.3 and 3.5), and end by providing a formula for the dimension of Aχ over K (Theorem 3.6). Some of the results of this paper appear in the Ph.D. dissertation [Tu] of the second author written under the direction of the first author. We thank the referee for some useful suggestions. 1. Notation and background Let G be a fixed subgroup of the symmetric group Σr. For the general results of this section and the next, the field K can be more general than the field of complex numbers. We assume only that K is an infinite field of characteristic not a divisor of jGj. Fix a positive integer n. The vector space E = Kn is acted on naturally (from the left) by the semigroup M = Matn(K) of n × n matrices over K and is therefore a KM- module, where KM is the semigroup algebra of M over K. The group G acts from the right on M r = M × · · · × M (r factors) by r mσ = (mσ(1); mσ(2); : : : ; mσ(r))(m 2 M ; σ 2 G). The semigroup M r acts on E⊗r = E ⊗ · · · ⊗ E (r factors) by mv = r ⊗r m1v1 ⊗ · · · ⊗ mrvr (m 2 M ; v = v1 ⊗ · · · ⊗ vr 2 E ). This action extends linearly to make E⊗r a KM r-module. For each 1 ≤ α ≤ n, let eα 2 E be the n-tuple with β-entry δαβ (Kronecker r ⊗r delta). Let I = fi = (i1; : : : ; ir) 2 Z j 1 ≤ ij ≤ n for all jg. The space E has basis fei j i 2 Ig, where ei := ei1 ⊗ · · · ⊗ eir . r Qr For m 2 M and i; j 2 I, define mi;j = α=1(mα)iαjα . 1.1 Lemma. Let m 2 M r. P (i) mej = i mi;jei (j 2 I). (ii)( mσ)i;j = miσ−1;jσ−1 (σ 2 G; i; j 2 I). Proof. (i) For all j 2 I, we have n ! n ! X X mej = m1ej1 ⊗ · · · ⊗ mrejr = (m1)i1j1 ei1 ⊗ · · · ⊗ (mr)irjr eir i1=1 ir=1 r X Y X = (mα)iαjα ei1 ⊗ · · · ⊗ eir = mi;jei: i2I α=1 i2I (ii) For all σ 2 G and i; j 2 I, we have r Y Y (mσ) = (m ) = (m ) i;j σ(α) iαjα α iσ−1(α)jσ−1(α) α=1 α Y −1 −1 = (mα)(iσ )α(jσ )α α = miσ−1;jσ−1 ; COEFFICIENT COALGEBRA OF SYMMETRIZED TENSOR SPACE 3 r r Let Γ = GLn(K) ⊆ M and put Γ = Γ×· · ·×Γ ⊆ M . The right action of G on M r induces an action on the group algebra KΓr of Γr over K. Denote by R the subalgebra of KΓr consisting of those elements fixed by G: R = (KΓr)G = fκ 2 KΓr j κσ = κ for all σ 2 Gg: The set fg¯ j g 2 Γrg is a K-basis for R, where 1 X g¯ = gσ: jGj σ2G Let V be an R-module with finite K-basis fvj j j 2 Jg. For κ 2 R and j 2 J, we have X κvj = αij(κ)vi i2J ∗ for some αij 2 R = HomK (R; K) (dual space of R). The K-linear span of the set fαij j i; j 2 Jg is the coefficient space of the R-module V , denoted cf(V ). This space is independent of the choice of basis for V . If RK (R) is the representative K-bialgebra of the multiplicative semigroup R, then V is a right RK (R)-comodule with structure map : V ! V ⊗ P RK (R) given by (vj) = i2J vi ⊗ αij. We have X 4(αij) = αik ⊗ αkj; k (αij) = δij; so that cf(V ) is a subcoalgebra of RK (R)[Ab, p. 125]. Let ρ : R ! EndK (V ) be the representation afforded by the R-module V . 1.2 Lemma. (i) For κ 2 R, we have κ 2 ker ρ if and only if f(κ) = 0 for all f 2 cf(V ). (ii) For f 2 R∗, we have f 2 cf(V ) if and only if f(κ) = 0 for all κ 2 ker ρ. Proof. See proof of [Ma, 2.2.1]. 1.3 Theorem. The map : im ρ ! cf(V )∗ given by (ρ(κ))(c) = c(κ) is a K-isomorphism. Proof. The bilinear map im ρ × cf(V ) ! K given by hρ(κ); ci = c(κ) is well defined and nondegenerate by Lemma 1.2. Since the spaces im ρ and cf(V ) are finite dimensional, the induced map : im ρ ! cf(V )∗ is a K- isomorphism. 4 RANDALL R. HOLMES DAVID P. TURNER 2. Generalized coefficient coalgebra and Schur algebra We continue to assume that K is an infinite field of characteristic not a divisor of jGj. Recall that R is the subalgebra of KΓr consisting of those elements fixed by G. For i; j 2 I define ci;j : R ! K by putting 1 X c (¯g) = (gσ) i;j jGj i;j σ2G (g 2 Γr) and extending linearly to R. 2.1 Lemma. The function ci;j is well-defined. Proof. Let g; h 2 Γr and assume thatg ¯ = h¯. Then h = gτ for some τ 2 G, so 1 X 1 X c (h¯) = (hσ) = (gτσ) = c (¯g): i;j jGj i;j jGj i;j i;j σ2G σ2G 2.2 Lemma. For κ 2 R and j 2 I, X κej = ci;j(κ)ei: i2I Proof. Let g 2 Γr. It suffices to establish the equality in the case κ =g ¯. For j 2 I we have 1 X 1 X X X ge¯ = (gσ)e = (gσ) e = c (¯g)e ; j jGj j jGj i;j i i;j i σ2G σ2G i2I i2I where the second equality uses Lemma 1.1. The group G acts on I from the right by iσ = (iσ(1); iσ(2); : : : ; iσ(r)). In turn, G acts on I × I diagonally: (i; j)σ = (iσ; jσ). For i; j; k; l 2 I, put (i; j) ∼ (k; l) if (k; l) = (i; j)σ = (iσ; jσ) for some σ 2 G. 2.3 Lemma. ci;j = ck;l if and only if (i; j) ∼ (k; l). r Proof. For i; j 2 I, definec ^i;j : M ! K by 1 X c^ (m) = (mσ) ; i;j jGj i;j σ2G r and note that for g 2 Γ we havec ^i;j(g) = ci;j(¯g). Let i; j; k; l 2 I and assume that ci;j = ck;l. Thenc ^i;j(g) =c ^k;l(g) for each r r r g 2 Γ and, since Γ is Zariski dense in M , it follows thatc ^i;j =c ^k;l.