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So far, the representation theory of Schur algebras has developed very well and many derivatives appeared, such as q-Schur algebras, infinitesimal Schur algebras, Borel-Schur algebras and so on. In particular, the representation type of Schur algebras is completely determined by various authors, including Erdmann [Er], Xi [Xi], Doty-Nakano [DN] and Doty-Erdmann-Martin-Nakano [DEMN]. We summarize these results in Proposition 3.1. Let n, r be two positive integers and V an n-dimensional vector space over F. We ⊗r denote by V the r-fold tensor product V ⊗F V ⊗F ···⊗F V . Then, the symmetric group ⊗r Gr has a natural action (by permutation) on V which makes it to be a module over ⊗r the group algebra FGr of Gr. Then, the endomorphism ring EndFGr (V ) is called the Schur algebra (see [Ma, Section 2]) and we denote it by S(n, r). Our initial motivation is to check whether τ-tilting finiteness and representation- finiteness coincide in the class of Schur algebras. It is clear that representation-finite Schur algebras are τ-tilting finite. Moreover, we have already determined the number of pairwise non-isomorphic basic support τ-tilting modules blockwise in Theorem 3.2. Therefore, we start with the tame cases. Theorem 1.1 (Theorem 3.4). If the Schur algebra S(n, r) is tame, it is τ-tilting finite. In order to prove the above result, we may show that any block algebra of tame Schur algebras is τ-tilting finite. Let S(n, r) be a tame Schur algebra and A := eS(n, r)e an indecomposable block algebra with a central idempotent e of S(n, r). It is known from [Er] and [DEMN, Section 5] that the block algebra A is Morita equivalent to one of F,
A2, D3, D4, R4 and H4 (see Section 3 for the definitions). Then, we have Theorem 1.2 (Theorem 3.2 and Lemma 3.3). Let sτ-tilt A be the set of pairwise non- isomorphic basic support τ-tilting A-modules. Then,
A A2 D3 D4 R4 H4 . #sτ-tilt A 6 28 114 88 96 Since the above results give a negative answer to our initial motivation, we next ask whether all wild Schur algebras are τ-tilting infinite. There are a few cases in (⋆) that we currently do not have a suitable method to deal with. Since the number of pairwise non-isomorphic basic support τ-tilting modules for them is huge, we cannot confirm the τ-tilting finiteness of these exceptions by direct calculation. So, we leave these cases here and expect to find a new reduction method (for example, Conjecture 4.13) to show the τ-tilting finiteness. p =2, n =2,r =8, 17, 19; p =2, n =3,r = 4; (⋆) p =2, n > 5,r = 5; p > 5, n =2,p2 6 r 6 p2 + p − 1. 2 Now, we are able to determine the τ-tilting finiteness for most wild Schur algebras.
Theorem 1.3 (Table 1, Table 2 and Table 3). Let p be the characteristic of F. Except for the cases in (⋆), the wild Schur algebra S(n, r) is τ-tilting finite if and only if p =2, n =2,r =6, 13, 15 or n =3,r =5 or n =4,r =4.
We have the following answers for our motivations.
Corollary 1.4. Let p =3. Then, S(n, r) is τ-tilting infinite if and only if it is wild.
Corollary 1.5. Let p > 5. Except for the cases in (⋆), S(n, r) is τ-tilting finite if and only if it is representation-finite.
Actually, except for the cases in (⋆), we have observed from the proof process that the Schur algebra S(n, r) is τ-tilting infinite if and only if it has one of τ-tilting infinite
quivers Q1, Q2 and Q3 (see Lemma 2.15) as a subquiver. This leads us to conjecture that
all cases in (⋆) are τ-tilting finite, because all the cases in (⋆) do not have one of Q1, Q2
and Q3 as a subquiver. This paper is organized as follows. In Section 2, we first recall some basic materials on τ-tilting theory and the Schur algebra S(n, r). Then, we give several reduction theorems on the τ-tilting finiteness of Schur algebras, such that we only need to consider small n and r. At the end of the section, we explain our strategy to prove S(n, r) to be τ-tilting infinite. In Section 3, we show that all tame Schur algebras are τ-tilting finite. In Section 4, we determine the τ-tilting finiteness of most wild Schur algebras.
Acknowledgements. This paper is inspired by another on-going joint work with Susumu Ariki, Ryoichi Kase, Kengo Miyamoto and Euiyong Park, where the authors are using τ-tilting theory to classify τ-tilting finite block algebras of Hecke algebras of classical type. I am very grateful to them, especially, Prof. Ariki, for giving me a lot of kind guidance in writing this paper. I am also very grateful to the anonymous referee for giving me many useful suggestions.
2 Preliminaries
We recall that any finite-dimensional basic algebra A over an algebraically closed field F is isomorphic to a bound quiver algebra FQ/I, where FQ is the path algebra of the
Gabriel quiver Q = QA of A and I is an admissible ideal of FQ. We refer to [ASS] for more background materials on quiver and representation theory of algebras.
2.1 τ-tilting theory
In this subsection, we review some background of τ-tilting theory. However, we only review the background needed for this paper, so that many aspects related to τ-tilting theory are omitted. One may look at [AIR], [Asa] and [AIRRT] for more materials.
3 Let mod A be the category of finitely generated right A-modules and proj A the category of finitely generated projective right A-modules. For any M ∈ mod A, we denote by add(M) (respectively, Fac(M)) the full subcategory of mod A whose objects are direct summands (respectively, factor modules) of finite direct sums of copies of M. We denote by Aop the opposite algebra of A and by |M| the number of isomorphism classes of indecomposable direct summands of M. In particular, we often describe A- modules via their composition factors. For example, each simple A-module Si is written S1 1 as i, and then S2 ≃ 2 is an indecomposable A-module M with a unique simple submodule
S2 such that M/S2 = S1. We recall that the Nakayama functor ν := D(−)∗ is induced by the dualities
op ∗ op D := HomF(−, F): mod A ↔ mod A and (−) := HomA(−, A): proj A ↔ proj A . Then, for any M ∈ mod A with a minimal projective presentation
f1 f0 P1 −→ P0 −→ M −→ 0, the Auslander-Reiten translation τM is defined by the exact sequence:
νf1 0 −→ τM −→ νP1 −→ νP0. Definition 2.1. ([AIR, Definition 0.1]) Let M ∈ mod A.
(1) M is called τ-rigid if HomA(M,τM) = 0. (2) M is called τ-tilting if M is τ-rigid and |M| = |A|. (3) M is called support τ-tilting if there exists an idempotent e of A such that M is a τ-tilting (A/AeA)-module. In the case of support τ-tilting modules, we may correspondingly define support τ- tilting pairs. For a pair (M,P ) with M ∈ mod A and P ∈ proj A, it is called a support
τ-tilting pair if M is τ-rigid, HomA(P, M) = 0 and |M| + |P | = |A|. Obviously, (M,P ) is a support τ-tilting pair if and only if M is a τ-tilting (A/AeA)-module and P = eA. Definition 2.2. An algebra A is called τ-tilting finite if it has only finitely many pairwise non-isomorphic basic τ-tilting modules. Otherwise, A is called τ-tilting infinite. We denote by τ-rigid A (respectively, sτ-tilt A) the set of isomorphism classes of indecomposable τ-rigid (respectively, basic support τ-tilting) A-modules. It is known from [AIR, Theorem 0.2] that any τ-rigid A-module is a direct summand of some τ- tilting A-modules. Moreover, we have the following statement. Proposition 2.3. ([DIJ, Corollary 2.9]) An algebra A is τ-tilting finite if and only if one of (equivalently, any of) the sets τ-rigid A and sτ-tilt A is a finite set. We recall the concept of left mutation, which is the core concept of τ-tilting theory. Before doing this, we need the definition of minimal left approximation. Let C be an additive category and X,Y objects of C. A morphism f : X → Z with Z ∈ add(Y ) is called a minimal left add(Y )-approximation of X if it satisfies:
4 • every h ∈ HomC(Z,Z) that satisfies h ◦ f = f is an automorphism, ′ ′ ′ ′ • HomC(f,Z ): HomC(Z,Z ) → HomC(X,Z ) is surjective for any Z ∈ add(Y ), where add(Y ) is the category of all direct summands of finite direct sums of copies of Y . Definition-Theorem 2.4. ([AIR, Definition 2.19, Theorem 2.30]) Let T = M ⊕ N be a basic support τ-tilting A-module with an indecomposable direct summand M satisfying M∈ / Fac(N). We take an exact sequence with a minimal left add(N)-approximation f: f M −→ N ′ −→ coker f −→ 0.
− We call µM (T ) := (coker f) ⊕ N the left mutation of T with respect to M, which is again + a basic support τ-tilting A-module. (The right mutation µM (T ) can be defined dually.)
We may construct a directed graph H(sτ-tilt A) by drawing an arrow from T1 to T2
if T2 is a left mutation of T1. On the other hand, we can regard sτ-tilt A as a poset with respect to a partial order ≤. For any M, N ∈ sτ-tilt A, let (M,P ) and (N, Q) be their corresponding support τ-tilting pairs, respectively. We say that N ≤ M if
Fac(N) ⊆ Fac(M) or, equivalently, HomA(N,τM) = 0 and add(P ) ⊆ add(Q). Then, H(sτ-tilt A) is exactly the Hasse quiver on the poset sτ-tilt A, see [AIR, Corollary 2.34]. The following statement implies that an algebra A is τ-tilting finite if we can find a finite connected component of H(sτ-tilt A). Proposition 2.5. ([AIR, Corollary 2.38]) If the Hasse quiver H(sτ-tilt A) contains a finite connected component, it exhausts all support τ-tilting A-modules.
∗ According to the duality (−) = HomA(−, A), we have Proposition 2.6. ([AIR, Theorem 2.14]) There exists a poset anti-isomorphism between sτ-tilt A and sτ-tilt Aop. We may give an example to illustrate the constructions above. α F / Example 2.7. Let A := (1 o 2 )/ < αβ,βα >. We denote by S1,S2 the simple β A-modules and by P1,P2 the indecomposable projective A-modules. We take an exact sequence with a minimal left add(P1)-approximation f of P2:
e2 α· e1 β −→ α −→ coker f −→ 0. − − Then, coker f = S1 and µP2 (A) = P1 ⊕ S1. Similarly, µP1 (A) = S2 ⊕ P2 and we can compute the left mutations step by step, so that the Hasse quiver H(sτ-tilt A) is > / > 3 P1 ⊕ S1 S1 ❙❙ > ❣❣❣❣❣❣ ❙❙❙❙ ❣❣ ❙❙) P1 ⊕ P2 > > ❦5 0 , ❲❲❲❲❲❲ ❦❦❦ ❲❲+ > ❦❦❦❦ S2 ⊕ P2 / S2 where > is the partial order on sτ-tilt A. Then, we have
• τ-rigid A = {P1,P2,S1,S2, 0} and sτ-tilt A = {P1 ⊕ P2,P1 ⊕ S1,S2 ⊕ P2,S1,S2, 0}.
In particular, P1 ⊕ P2,P1 ⊕ S1 and S2 ⊕ P2 are τ-tilting A-modules.
• (S1,P2) and (S2,P1) are support τ-tilting pairs.
5 2.2 The symmetric group and the Schur algebra
We refer to some textbooks, such as [Ja], [Ma] and [Sa], for the representation theory of the symmetric group and the Schur algebra.
Let r be a natural number and λ := (λ1,λ2,...) a sequence of non-negative integers.
We call λ a partition of r if i∈N λi = r with λ1 > λ2 > ... > 0, and the elements λi are called parts of λ. If there exists an n ∈ N such that λ = 0 for all i > n, we denote λ by P i (λ1,λ2,...,λn) and call it a partition of r with at most n parts. We denote by Ω(r) the set of all partitions of r and by Ω(n, r) the set of all partitions of r with at most n parts. It is well-known that Ω(r) admits the dominance order ⊲ and the lexicographic order >, we omit the definitions.
We denote by Gr the symmetric group on r symbols and by FGr the group algebra
of Gr. Each partition λ of r gives a Young subgroup Gλ of Gr defined as
Gλ := G{1,2,...,λ1} × G{λ1+1,λ1+2,...,λ1+λ2} ×···× G{λ1+···+λn−1+1,λ1+···+λn−1+2,...,r}.
F λ Gr Then, the permutation Gr-module M is 1Gλ ↑ , where 1Gλ denotes the trivial module for Gλ and ↑ denotes induction. ⊗r Let S(n, r) = EndFGr (V ) be the Schur algebra. It is well-known (e.g., see [Ve, Section 1.6]) that M λ with λ ∈ Ω(n, r) can be regarded as direct summands of V ⊗r. Therefore, we have the following algebra isomorphism,
λ S(n, r) ≃ EndFGr nλM , λ∈Ω(n,r) ! L where 1 6 nλ ∈ N is the number of compositions of r with at most n parts which are rearrangement of λ.
2.2.1 Specht modules
We follow the conventions in [Ja] for the constructions of Specht modules. We may regard a partition λ of r as a box-diagram [λ] of which the i-th row contains
λi-boxes. For a prime number p, a partition λ (or a diagram [λ]) is called p-regular if no p rows of λ have the same length. Otherwise, λ (or [λ]) is called p-singular. Let λ be a partition of r. A λ-tableau t is obtained from [λ] by filling the boxes by numbers {1, 2,...,r} without repetition. In fact, t is a bijection between the boxes in [λ] and the numbers in {1, 2,...,r}. For any σ ∈ Gr, we define an action t · σ := t ◦ σ by the composition of the bijection t and the permutation σ. Then, the column stabilizer of
a λ-tableau t is defined as the subgroup Ct of Gr consisting of permutations preserving
the numbers in each column of t. Similarly, the row stabilizer of t is the subgroup Rt consisting of permutations preserving the numbers in each row of t. Let t, t′ be two λ-tableaux. We may define a row-equivalence relation t ∼ t′ if t′ = t·σ
for σ ∈ Rt. We denote by {t} the equivalence class of t under ∼ and call it a λ-tabloid.
We also define a Gr-action on a λ-tabloid {t} by {t}· σ := {t · σ} for any σ ∈ Gr, and this
6 action is well-defined. Then, the λ-polytabloid et associated with a λ-tableau t is defined by et := {t}· κt, where κt := sgn(σ)σ is the signed column sum. σ∈Ct λ Let M be the permutationP module of FGr corresponding to a partition λ. It is known that M λ is spanned by all λ-tabloids. Then, we call the submodule Sλ of M λ spanned by all λ-polytabloids the Specht module corresponding to λ. If F is a field of characteristic zero, the set Sλ | λ ∈ Ω(r) is a complete set of pairwise non-isomorphic F F λ simple Gr-modules. If is a field of prime characteristic p, each Specht module S with λ being p-regular has a unique (up to isomorphism) simple top and we may denote it by Dλ. Then, the set Dλ | λ ∈ Ω(r),λ is p-regular is a complete set of pairwise F non-isomorphic simple Gr-modules. (In the case of a p -singular partition µ, all of the composition factors of Sµ are Dλ such that λ is a p-regular partition which dominates µ.) Let F be a field of prime characteristic p. The decomposition number [Sλ : Dµ] provides how many times each simple module Dµ occurs as a composition factor of the λ Specht module S . If we run all partitions of r, we get the decomposition matrix of FGr. We recall from [Ja, Corollary 12.3] that the decomposition matrix has the following form.
Dµ,µ p-regular
1 z ∗ 1 }| O { ∗ ∗ 1 Sλ,λ p-regular . . . . . . . .. ∗ ∗ ∗ ... 1 −−− − − ∗ ∗ ∗ ... ∗ Sλ,λ p-singular ∗ ∗ ∗ ... ∗ ( Although it is still an unsolved problem to determine the decomposition matrix of FGr, James [Ja] provides us with enough materials to write this paper.
2.2.2 Young modules
We recall that a permutation module M λ over F is liftable by a p-modular system and therefore, it has an associated ordinary character ch M λ. Let χλ be the ordinary character corresponding to Specht module Sλ over a field of characteristic zero. Then, χλ is a constituent of ch M λ and χµ(µ =6 λ) is a constituent of ch M λ if and only if µ⊲λ. λ n N We decompose M into some indecomposable direct summands ⊕i=1Yi for n ∈ . Obviously, each summand Yi is also liftable and has an associated ordinary character ch Yi. Then, the unique (up to isomorphism) direct summand Yi which the ordinary λ character χ occurs in ch Yi, is called the Young module corresponding to λ and we denote it by Y λ. It is well-known that the Young module Y λ has a Specht filtration given λ by Y = Z1 ⊇ Z2 ⊇ ···⊇ Zk = 0 for some k ∈ N, where each Zi/Zi+1 is isomorphic
7 to a Specht module Sµ with µ D λ. Moreover, the Young module Y λ is self-dual, that λ λ λ is, Y ≃ D(Y ) with respect to D = HomF(−, F). In fact, D(Y ) becomes a right −1 λ λ FGr-module via (f · σ)(x) := f(xσ ) for f ∈ D(Y ), σ ∈ Gr and x ∈ Y . Proposition 2.8. ([Ma, Section 4.6]) The set Y λ | λ ∈ Ω(n, r) is a complete set of pairwise non-isomorphic Young modules which occurs as indecomposable direct summands of permutation modules in M λ | λ ∈ Ω(n, r) . It is worth mentioning that if λ is a partition with at most two parts, Henke [He] provided a formula to calculate ch Y λ. We recall these constructions as follows. k Let p be a prime number. There is a p-adic decomposition s = s>0 skp for any non-negative integer s. Now, let s, t be two non-negative integers, we define a function P p − 1 − s f(s, t)= k , k∈{0}∪N p − 1 − tk ! m Q where we set n = 0 if m < n. Moreover, we define 1 if f(2t, s + t)=1, 1 if f(2t +1,s + t +1)=1, g(s, t) := and h(s, t) := (0 otherwise. (0 otherwise. Theorem 2.9. ([He, Section 5.2]) Let (r −k,k) be a partition with a non-negative integer k and ch Y (r−k,k) the associated ordinary character of Y (r−k,k). (1) If r is even, then
r 2 (r−k,k) r r (r−i,i) ch Y = g( 2 − i, 2 − k)χ . i=0 (2) If r is odd, then P
r [ 2 ] (r−k,k) r r (r−i,i) ch Y = h([ 2 ] − i, [ 2 ] − k)χ , i=0
r P r where [ 2 ] is the greatest integer less than or equal to 2 . We now explain how to construct the basic algebra of the Schur algebra S(n, r). Let
B be a block of the group algebra FGr labeled by a p-core ω. It is well-known that a partition λ belongs to B if and only if λ has the same p-core ω. Then, we define
λ SB := EndFGr Y λ∈B∩Ω(n,r) ! L and the basic algebra of S(n, r) is SB, where the sum is taken over all blocks of FGr. Moreover, S is a direct sum of blocks of the basic algebra of S(n, r). We remark that if B L we consider the Young modules Y λ for partitions λ of r with at most n parts, the Specht modules Sµ in the Specht filtration of Y λ and the composition factors Dµ which appear in Y λ are also corresponding to the partitions with at most n parts. (The reason is that [Sλ : Dµ] =06 ⇒ µ ☎ λ.) We may give an example to illustrate our constructions above.
8 Example 2.10. We look at the Schur algebra S(2, 11) over p = 2. Let B1 be the principal
block of FG11 and B2 the block of FG11 labeled by 2-core (2, 1). Then, we may find in λ µ [Ja] that the parts of the decomposition matrix [S : D ] for the partitions in B1 and B2 with at most two parts are
(11) 1 (10, 1) 1 B1 : (9, 2) 0 1 , B2 : (8, 3) 1 1 . (7, 4) 1 0 1 (6, 5) 0 1 1 We determine SB2 as follows. By using the formula given in Theorem 2.9, we have
ch Y (10,1) = χ(10,1), ch Y (8,3) = χ(10,1) + χ(8,3), ch Y (6,5) = χ(10,1) + χ(8,3) + χ(6,5).
Similar to the proof of [Er, Lemma 4.4], we may read off the Specht filtration of Y λ from the formula, and the composition factors of Young modules are {D(10,1),D(8,3),D(6,5)}. It is obvious that Y (10,1) = S(10,1) = D(10,1). By the decomposition matrix above, the Specht module S(8,3) has composition factors {D(10,1),D(8,3)}. Since the top of S(8,3) is D(8,3) and S(8,3) is a submodule of Y (8,3), the simple module D(10,1) is in the socle of Y (8,3). We deduce the radical series of Y (8,3) by using the self-duality of Young modules, that is,
D(10,1) S(10,1) Y (8,3) = = D(8,3) . S(8,3) D(10,1)
Similarly, the simple module D(10,1) appears in the top of Y (6,5) because Y (6,5) has Specht filtration whose top is S(10,1). As(6, 5) is 2-regular, the top of S(6,5) is D(6,5) and the socle of S(6,5) is D(8,3). Since S(6,5) is the bottom Specht module, it implies that D(8,3) appears in the socle of Y (6,5). By the self-duality of Young modules, we deduce that
D(8,3) D(10,1) ❏❏ ✆✆ ❏ ✆✆ Y (6,5) = ✆✆ (6,5) ✆✆ . ✆✆ D ❏ ✆✆ ✆✆ ❏❏ ✆✆ D(10,1) D(8,3) (10,1) (8,3) (6,5) F Thus, SB2 = EndFG11 (Y ⊕ Y ⊕ Y ) is isomorphic to Q/I with
α1 α2 / / Q : (10, 1) o (6, 5) o (8, 3) and I : α1β1, β2α2,α1α2β2,α2β2β1 , β1 β2 D E where we replace each vertex in Q by λ associated with the Young module Y λ. F F Similarly, one may find that SB1 is isomorphic to Q/I ⊕ , where
α1 / Q : (11) o (7, 4) and I : α1β1 . β1 D E
Therefore, the basic algebra of S(2, 11) over p = 2 is SB1 ⊕ SB2 .
9 2.3 Reduction theorems
We first recall some reduction theorems for the τ-tilting finiteness of an algebra A. Proposition 2.11. ([AIRRT, Theorem 5.12], [DIJ, Theorem 4.2]) If A is τ-tilting finite, (1) the quotient algebra A/I is τ-tilting finite for any two-sided ideal I of A, (2) the idempotent truncation eAe is τ-tilting finite for any idempotent e of A. In addition, Eisele, Janssens and Raedschelders [EJR] provided us with a powerful reduction theorem, as shown below. Proposition 2.12. ([EJR, Theorem 1]) Let I be a two-sided ideal generated by central elements which are contained in the Jacobson radical rad A of A. Then, there exists a poset isomorphism between sτ-tilt A and sτ-tilt (A/I). Next, we focus on the Schur algebra S(n, r). We give two useful reduction theorems which will allow us to simplify the general problems to the cases with small n and r. We point out that S(n, r) with n>r is always Morita equivalent to S(r, r). Lemma 2.13. If S(n, r) is τ-tilting infinite, then so is S(N,r) for any N > n.
Proof. Let S be the basic algebra of S(N,r). For each λ ∈ Ω(n, r), we define eλ to be λ the projector to Y and take the sum e := eλ over all partitions in Ω(n, r). Then, the idempotent truncation is P
λ λ eSe = eEndFGr Y e = EndFGr Y . λ∈Ω(N,r) ! λ∈Ω(n,r) ! L L This implies that the basic algebra of S(n, r) is an idempotent truncation of S(N,r) for any N > n. Thus, the statement follows from Proposition 2.11.
We recall that the coordinate function cij : GLn(F) → F is defined by cij(g) = gij
for all g = [gij] ∈ GLn(F), where i, j ∈ {1, 2,...,n}. Then, we denote by A(n, r) the coalgebra generated by the homogeneous polynomials of total degree r in cij. In fact, the Schur algebra S(n, r) is just the dual of A(n, r). Lemma 2.14. If S(n, r) is τ-tilting infinite, then so is S(n, n + r). Proof. It has been proved in [Er] that S(n, r) is a quotient of S(n, n+r). For convenience, we recall the proof as follows. Let I = I(n, n + r) be the set of maps α : {1, 2,...,n + r}→{1, 2,...,n}
with right Gn+r-action. Then, S(n, n + r) has a basis {ξα,β | (α, β) ∈ (I × I)/Gn+r}: the
dual basis of cα(1)β(1)cα(2)β(2) ...cα(n+r)β(n+r) ∈ A(n, n + r). Then, the elements ξα := ξα,α form a set of orthogonal idempotents for S(n, n + r) whose sum is the identity. Note
that Ω(n, n + r) ⊆ I is the set of representatives of Gn+r-orbits. Let e = ξλ be the idempotent of S(n, n + r), where the sum is taken over all λ ∈ Ω(n, n + r) such that P λn = 0. Then, by using det (cij), we may prove
10 S(n, n + r)/S(n, n + r)eS(n, n + r) ≃ S(n, r). Therefore, the statement follows from Proposition 2.11.
2.4 Strategy on τ-tilting infinite Schur algebras
Let A := FQ/I be an algebra presented by a quiver Q and an admissible ideal I. We call Q a τ-tilting infinite quiver if A/rad2A is τ-tilting infinite. For example, the / Kronecker quiver Q : ◦ / ◦ is a τ-tilting infinite quiver. Then, the following lemma provides us with three τ-tilting infinite quivers.
Lemma 2.15. The following quivers Q1, Q2 and Q3 are τ-tilting infinite quivers.
/ / / Q1 : ◦ o ◦ , Q2 : ◦ ❈ = ◦ , Q3 : ◦ o ◦ o ◦ . O O a❈❈❈❈ ④④④ O ❈❈❈❈ ④④④④ ❈❈❈❈ ④④④④ ❈ ❈ ④④ ④ / ❈/ ! }④④ / / / ◦ o ◦ ◦ o ◦ o ◦ ◦ o ◦ o ◦ Proof. We look at the following subquivers. ◦ / ◦ ◦ ◦ ◦ / ◦o ◦ O ❈ O ❈❈ ④④ Q′ Q′ ❈❈ ④④ Q′ 1 : , 2 : ❈❈ ④④ , 3 : . ❈! }④④ ◦o ◦ ◦ / ◦o ◦ ◦o ◦ / ◦ ′ F Since the path algebra of Qi for i =1, 2, 3 is a quotient algebra of A = Q/I if Q = Qi, and all of these path algebras are τ-tilting infinite as mentioned in [W2], we conclude 2 that A/rad A is τ-tilting infinite if Q = Q1, Q2, Q3 by Proposition 2.11.
We remark that Adachi [Ada] (and Aoki [Ao]) provided a handy criteria for the τ- tilting finiteness of radical square zero algebras, that is, for any algebra A, the quotient A/rad2A is τ-tilting finite if and only if every single subquiver of the separated quiver for A/rad2A is a disjoint union of Dynkin quivers. This also gives a proof of Lemma 2.15. We mention that in order to show that S(n, r) is τ-tilting infinite, it suffices to find a block algebra of S(n, r) which is Morita equivalent to FQ/I with a τ-tilting infinite subquiver in Q. Then, the advantage is that it is not necessary to find the explicit relations in I. As we mentioned in the introduction, except for the cases in (⋆), S(n, r) is τ-tilting infinite if and only if it contains one of Q1, Q2 and Q3 as a subquiver. To find a τ-tilting infinite subquiver in S(2,r), it is worth mentioning Erdmann and
Henke’s method [EH1]. Let λ =(λ1,λ2) and µ =(µ1,µ2) be two partitions of r, we define s two non-negative integers s := λ1 − λ2 and t := µ1 − µ2. We denote by v the vertex in (λ1,λ2) the quiver of S(2,r) corresponding to the Young module Y with s = λ1 − λ2. Let n(vs, vt) be the number of arrows from vs to vt. Then, it is shown in [EH1, Theorem 3.1] that n(vs, vt) = n(vt, vs) and n(vs, vt) is either 0 or 1. We have the following recursive algorithm for computing n(vs, vt).
Lemma 2.16. ([EH1, Proposition 3.1]) Suppose that p is a prime number and s > t. Let ′ ′ ′ ′ s = s0 + ps and t = t0 + pt with 0 6 s0, t0 6 p − 1 and s , t > 0.
11 (1) If p =2, then
s′ t′ ′ ′ n(v , v ) if s0 = t0 =1 or s0 = t0 =0 and s ≡ t mod 2, s t ′ ′ n(v , v )= 1 if s0 = t0 =0, t +1= s 6≡ 0 mod 2, 0 otherwise. (2) If p> 2, then
s′ t′ n(v , v ) if s0 = t0, s t ′ ′ n(v , v )= 1 if s0 + t0 = p − 2, t +1= s 6≡ 0 mod p, 0 otherwise. 3 Representation-finite and tame Schur algebras
In this section, we show that all tame Schur algebras are τ-tilting finite. We first recall the complete classification of the representation type of Schur algebras. Note that some semi-simple cases are contained in the representation-finite cases. We may distinguish the semi-simple cases following [DN]. Namely, the Schur algebra S(n, r) is semi-simple if and only if p =0 or p>r or p =2, n =2,r = 3.
Proposition 3.1. ([Er, DEMN], Modified1) Let p be the characteristic of F. Then, the Schur algebra S(n, r) is representation-finite if and only if p = 2, n = 2,r = 5, 7 or p > 2, n =2,r
3.1 Representation-finite blocks
Erdmann [Er, Proposition 4.1] showed that each block A of a representation-finite
Schur algebra S(n, r) is Morita equivalent to Am := FQ/I for some m ∈ N, which is defined by the following quiver with relations.
α1 α2 αm−2 αm−1 / / / / Q : 1 o 2 o ··· o m − 1 o m , β1 β2 βm−2 βm−1 (3.1)
I : hα1β1,αiαi+1, βi+1βi, βiαi − αi+1βi+1 | 1 6 i 6 m − 2i .
Three years later after [Er], Donkin and Reiten [DR, Theorem 2.1] generalized this result to an arbitrary Schur algebra S(n, r), that is, each representation-finite block of Schur algebras is Morita equivalent to Am for some m ∈ N. We would like to determine the number of pairwise non-isomorphic basic support τ-tilting modules for a representation-finite block of Schur algebras.
1The Schur algebra S(2, 11) over p = 2 was proved to be wild in [DEMN], but this is an error caused by miscalculating the number of simples in each of the blocks of S(2, 11). After correcting the number, S(2, 11) becomes tame because we can apply [EH2, Theorem 13] to show that S(2, 9) and S(2, 11) have the same representation type. Thanks to Prof. Erdmann for explaining this.
12 2m Theorem 3.2. Let Am be the algebra defined above. Then, #sτ-tilt Am = m . Proof. Let Λm be the Brauer tree algebra whose Brauer tree is a straight line having m + 1 vertices and without exceptional vertex. Then, it is easy to check that Am is a quotient algebra of Λm modulo the two-sided ideal generated by α1β1. Since α1β1 is a 2m central element of Λm and #sτ-tilt Λm = m has been determined in [Ao, Theorem 5.6], we get the statement following Proposition 2.12.
3.2 Tame Schur algebras
The block algebras of tame Schur algebras are well-studied in [Er] and [DEMN]. In this subsection, we recall these constructions and show that tame Schur algebras are τ- tilting finite. We recall the following bound quiver algebras constructed in [Er], where the tameness for them is given in [DEMN, 5.5, 5.6, 5.7].
• Let D3 := FQ/I be the special biserial algebra given by
α1 α2 / / Q : 1 o 2 o 3 and I : α1β1, β2α2,α1α2β2,α2β2β1 . (3.2) β1 β2 D E
• Let D4 := FQ/I be the bound quiver algebra given by
α1 β3 α β ,α β ,α β ,α β ,α β ,α β , Q : 1 / 3 / 4 and I : 1 1 2 2 3 1 3 2 1 3 2 3 . (3.3) o O o α3 β1 * α1β2α2, β2α2β1, β2α2 − β3α3 + α2 β2 2
• Let R4 := FQ/I be the bound quiver algebra given by
α1 α2 α3 / / / α1β1,α1α2, β2β1, Q : ◦ o ◦ o ◦ o ◦ and I : . (3.4) β1 β2 β3 *α2β2 − β1α1,α3β3 − β2α2+
• Let H4 := FQ/I be the bound quiver algebra given by
α1 α3 α β ,α β ,α β ,α β ,α α , Q : ◦ / ◦ / ◦ and I : 1 1 1 2 2 1 2 2 1 3 . (3.5) o O o β1 β3 β3β1,α3β3 − β1α1 − β2α2 α2 β2 * + ◦
We remark that D3, D4, R4 and H4 are also tame blocks of some wild Schur algebras.
Lemma 3.3. The tame algebras D3, D4, R4 and H4 are τ-tilting finite. Moreover,
A D3 D4 R4 H4 . #sτ-tilt A 28 114 88 96
13 Proof. We often use Proposition 2.12 to reduce the direct calculation of left mutations. 1 (1) Since α2β2 and β2β1α1α2 are non-trivial central elements of D3, we may define
D3 := D3/<α2β2, β2β1α1α2 > so that #sτ-tilt D = #sτ-tilt D . Then, we determine the number #sτ-tilt D by 3 f 3 3 calculating the left mutations starting with D3. In fact, this is equivalent to finding the f f Hasse quiver H(sτ-tilt D3). We recall that the indecomposable projective D3-modules are f2 3 1 1 3 2 f P1 = 2 ,P2 = 2 ,P3 = 1 . f 3 3 2
Starting with the unique maximal τ-tilting module D3 ≃ P1 ⊕ P2 ⊕ P3, we take an exact sequence with a minimal left add(P2 ⊕ P3)-approximation f1 of P1: 2 f 1 f1 1 3 2 −→ 2 −→ coker f1 −→ 0. 3 3
2 − 2 Then, coker f1 = 3 and µP1 (D3) = 3 ⊕ P2 ⊕ P3. Next, we take an exact sequence with a 2 minimal left add(3 ⊕ P3)-approximation f2 of P2: f f2 2 P2 −→ 3 ⊕ P3 −→ coker f2 −→ 0,
3 − − 2 3 − − − 2 3 so that coker f2 = 2 and µP2 (µP1 (D3)) = 3 ⊕ 2 ⊕ P3. Similarly, µP3 (µP2 (µP1 (D3))) = 3 ⊕ 2. Then, by the calculation in Example 2.7, we have f f 3 3 ⊕ 2 3 2 3 3 ⊕ 2 0 2 ⊕ 2 2 3 .
In this way, one can calculate all possible left mutation sequences starting with D3 and ending at 0, so that the Hasse quiver H(sτ-tilt D3) is as follows. f ◦ ❑ / ◦ ❯❯❯ 9/ ◦ / ◦ / ◦ ✆B ❑❑ ❯❯❯ ss ✴ C ✴ ✆ ❑❑ ❯❯❯❯ ss f ✴✴ ✞✞ ✴✴ ✆ ❑❑❑ ❯❯s❯ss ✴ ✞ ✴ ✆✆ ❑% ss ❯❯❯❯ ✴ ✞✞ ✴ ✆✆ ◦ / ◦ 5* ◦ ✴ ✞ ✴✴ ✆ ❍❍ ❍❍ ❥❥❥ ❍❍ ✴✴✞✞ ✴ ✆✆ ❍❍ ❥❍❥❍❥ ❍❍ ✞✴ ✴ ✆ ❍❍ ❥❥❥❥ ❍❍ ❍❍ ✞ ✴ ✴ ✆ ❥❍❥❍❥ ❍❍ ❍❍ ✞✞ ✴ ✴✴ ❥❥❥❥ ❍$ ❍$ ❍$ ✞ ✴✴ ✴ D3 / ◦ / ◦ ◦ / ◦ ◦ ✴ ✴ ✵ ✴ ❍❍ ✈; ❍❍ ✈; ✴ ✴✴ ✵ ✴✴ ❍❍ ✈✈ ❍❍ ✈✈ ✴ ✴ ✵ ✴ ❍✈❍✈ ❍✈❍✈ ✴✴ ✴ . ✵✵ ✴ ✈✈ ❍❍ ✈✈ ❍❍ ✴ ✴ f ✵ ✴✴ ✈✈ ❍# ✈✈ ❍# ✵✵ ✴ ◦ ❏ ◦ ❏ / ◦ ❏ ✐✐4 ◦ / ◦ / 0 ✵ ✴✴ ✝B ❏❏ ❏❏ ❏❏✐✐✐ ✝B ✵ ✴ ✝ ❏❏❏ ❏❏❏ ✐✐✐✐❏❏❏ ✝ ✵✵ ✴✝✝ ❏❏ ✐❏✐❏✐✐ ❏❏ ✝✝ ✵ ✝✴✴ ❏% ✐✐✐✐ ❏% ❏% ✝ ✵ ✝ ✴ ◦ ❯❯ ◦ / ◦ ✝ ✵ ✝✝ ✴ ❯❯❯ s9 ❑❑❑ ✝✝ ✵ ✝ ✴ ss❯s❯❯❯ ❑❑ ✝ ✵ ✝ ✴✴ ss ❯❯❯ ❑❑ ✝ ✵ ✝✝ ss ❯❯❯❯ ❑❑ ✝✝ ◦ / ◦ / ◦ s ❯/* ◦ %/ ◦
Hence, we deduce that D3 is τ-tilting finite and #sτ-tilt D3 = 28.
(2) Since β1α1 and β2α2 + β3α3 are non-trivial central elements of R4, we define
R4 := R4/ < β1α1, β2α2 + β3α3 >
1We use GAP: a System for Computational Discrete Algebra to compute the central elements. f
14 and then, #sτ-tilt R4 = #sτ-tilt R4. Instead of direct calculation, we point out that
R4 is a representation-finite string algebra and H(sτ-tilt R4) can be constructed by the f String Applet [Ge]. Thus, we deduce that #sτ-tilt R4 = 88. f f (3) Since the non-trivial central elements of D4 are β2α2, α3β3 and α2β1α1β2, we define
D4 := D4/ < β2α2,α3β3,α2β1α1β2 > so that #sτ-tilt D = #sτ-tilt D . We explain our strategy for computing #sτ-tilt D . 4 f 4 4 We denote by as(D4) the number of pairwise non-isomorphic basic support τ-tilting f f D4-modules M with support-rank s (i.e., |M| = s) for 0 6 s 6 4. Then, we have f n f #sτ-tilt D4 = as(D4). s=0 P Since the support τ-tilting D4-module withf support-rankf 0 is unique, we have a0(D4) = 1.
Since each simple D4-module Si is a D4/< 1 − ei >-module and D4/< 1 − ei >≃ F, we f f observe that the support τ-tilting D4-modules with support-rank 1 are exactly the simple f f f D4-modules. Then, a1(D4)= |D4| = 4. f Let M be a support τ-tilting D4-module with support-rank 2, and with supports ei f f f and ej (i =6 j). Then, M becomes a τ-tilting D4/J-module with J :=< 1 − ei − ej >. We f denote by bi,j the number of τ-tilting D4/J-modules. For example, if (i, j)=(1, 3), then f α1 F / D4/J = f1 o 3 / hα1β1i. β1
Note that β1α1 is a central elementf of D4/J, we may apply Proposition 2.12 and Example
2.7 to show that H(sτ-tilt D4/J) is displayed below, f • / ◦ ❯ ✐✐✐✐4 ❯❯❯❯ f ✐✐✐✐ ❯❯❯* • ❯❯❯ ✐4 ◦ , ❯❯❯❯ ✐✐✐✐ ❯* • / ◦ ✐✐✐ where we denote by • τ-tilting D4/J-modules and by ◦ other support τ-tilting (but not
τ-tilting) D4/J-modules. Hence, we deduce that b1,3 = 3. Similarly, we have f (i, j) (1, 2) (1, 4) (2, 3) (2, 4) (3, 4) f bi,j 1 1 3 1 3
This implies that a2(D4) = 12.
Let N be a support τ-tilting D4-module with support-rank 3. Then, N becomes a f τ-tilting D4/Lj-module with Lj :=< ej >, where ej is the only one non-zero primitive f idempotent satisfying Nej = 0. We denote by dj the number of τ-tilting D4/Lj-modules. For example,f if j = 4, then f α1 β2 F / / D4/L4 = 1 o 3 o 2 / hα1β1,α2β2, β2α2,α2β1α1β2i. β1 α2
Since D4/L4 fis isomorphic to D3, d4 = 17 by the calculation before. If j = 2, then
α1 β3 F / / f D4/L2 = 1 o f 3 o 4 / hα1β1,α3β1,α1β3, β3α3,α3β3i, β1 α3 f 15 and the Hasse quiver H(sτ-tilt D4/L2) is shown as follows by direct calculation.
• / • / • / ◦ / ◦ B ❑❑ ❯❯❯❯ s9 ✾ s9 ✾ ✆ ❑❑ f ❯s❯s❯❯ ✾ ss ✾ ✆✆ ❑❑❑ sss ❯❯❯❯ ✾✾sss ✾✾ ✆ ❑❑ ss ❯❯❯❯ ss✾ ✾ ✆✆ % s ❯* s ✾✾ ✾✾ ✆ • ❯❯❯❯ ✐✐✐4 ◦ ✾ ✾ ✆✆ ❯❯❯❯ ✐✐✐✐ ✾✾ ✾✾ ✆ ✐✐✐❯✐❯❯❯ ✾ ✾ ✆✆ ✐✐✐✐ ❯❯❯❯ ✾ ✾ • / • / ◦ ✐✐ ❯* ◦ / ◦ / ◦ ✾ ✾ ❯❯❯❯ ✐✐✐4 B ✾ ✾ ❯❯❯❯ ✐✐✐✐ ✆ ✾✾ ✾✾ ✐✐❯✐❯✐❯❯ ✆✆ ✾ ✾ ✐✐✐✐ ❯❯❯❯ ✆ ✾✾ ✾✾ ✐✐ ❯* ✆✆ ✾ ✾ s9 ◦ ❯❯❯❯ s9 ◦ ❑❑ ✆ ✾✾ s✾s✾ ❯❯❯❯ ss ❑❑ ✆✆ ✾ sss ✾ ❯❯❯❯sss ❑❑❑ ✆ ✾ ss ✾ ss❯❯❯❯ ❑❑ ✆✆ • s / • / • s ❯/* ◦ /% ◦
We deduce that d2 = 9. Similarly, we have d1 = 9 and d3 = 1. Therefore, a3(D4) = 36.
We may compute a4(D4) by hand, because the support τ-tilting D4-modules with f support-rank 4 are just τ-tilting D4-modules, which can be obtained by the left mutations f f starting with D4. See Appendix A for a complete list of τ-tilting D4-modules and one may f easily construct the part of H(sτ-tilt D4) consisting of all τ-tilting D4-modules. Then, f f a4(D4) = 61 and we conclude that #sτ-tilt D4 =1+4+12+36+61=114. f f (4) Since the non-trivial central elements of H4 are β1α1, β2α2 + β3α3 and β3β2α2α3, wef define f
H4 := H4/ < β1α1, β2α2 + β3α3, β3β2α2α3 > so that #sτ-tilt H = #sτ-tilt H . Similar to the case of D , one can compute the left 4 f 4 4 mutations starting with H4 to find all τ-tilting H4-modules and the number is 47. Then, f s tilt fs 012 3f 4 # τ- H4 a (H ) 1 4 12 32 47 96 s 4 f Also, the part of H(sτ-tilt Hf4) consisting of all τ-tilting H4-modules can be obtained.
Theorem 3.4. If the Schurf algebra S(n, r) is tame, it isfτ-tilting finite.
Proof. We have proved in Lemma 3.3 that D3, D4, R4 and H4 are τ-tilting finite. Now, it suffices to make clear that these are all the tame blocks of tame Schur algebras. By Proposition 3.1, it is enough to consider S(2,r) for r = 4, 9, 11 over p = 2, S(2,r) for r =9, 10, 11 and S(3,r) for r =7, 8 over p = 3. We have already shown in Example 2.10
that the basic algebra of S(2, 11) over p = 2 is isomorphic to D3 ⊕A2 ⊕ F. Then, the basic algebra of other tame Schur algebras can be found in [DEMN, Section 5]. We recall the result in [DEMN] as follows.
Let p = 2, the basic algebra of S(2, 4) is isomorphic to D3 and the basic algebra of
S(2, 9) is isomorphic to D3 ⊕F⊕F. Let p = 3, the basic algebra of S(2, 9) is isomorphic to
D4 ⊕ F, the basic algebra of S(2, 10) is isomorphic to D4 ⊕ F ⊕ F and the basic algebra of
S(2, 11) is isomorphic to D4⊕A2; the basic algebra of S(3, 7) is isomorphic to R4⊕A2 ⊕A2 and the basic algebra of S(3, 8) is isomorphic to R4 ⊕ H4 ⊕A2.
16 Table 1: The τ-tilting finite S(n, r) over p = 2. r 1 2 3 4 5 6 7 8 9 10 11 12 S(2,r) S F S T F W F W T W T W
r 13 14 15 16 17 18 19 20 21 22 23 ··· S(2,r) W W W W W W W W W W W ··· r 1 2 3 4 5 6 7 8 9 10 11 12 13 ··· n 3 S F F W W W W W W W W W W ··· 4 S F F W W W W W W W W W W ··· 5 S F F W W W W W W W W W W ··· 6 S F F W W W W W W W W W W ··· ......
4 Wild Schur Algebras
As we mentioned in the introduction, the cases in (⋆) have been distinguished and we will deal with them in the subsection 4.4. Now, we determine the τ-tilting finiteness for wild Schur algebras over various p as follows. In this section, we will use the decomposition λ µ matrix [S : D ] of FGr given in [Ja] without further notice.
4.1 The characteristic p =2
We assume in this subsection that the characteristic of F is 2. Then, the τ-tilting finiteness for S(n, r) is shown in Table 1 and the proof is divided into several propositions as displayed below. In particular, we claim that the color purple means τ-tilting finite, the color red means τ-tilting infinite, the capital letter S means semi-simple, the capital letter F means representation-finite, the capital letter T means tame and the capital letter W means wild. Proposition 4.1. Let p =2. Then, the wild Schur algebras S(2, 6), S(2, 13) and S(2, 15) are τ-tilting finite. Proof. We consider the Young modules Y λ for partitions λ with at most two parts. (1) The part of the decomposition matrix [Sλ : Dµ] for the partitions in the principal
block of FG6 with at most two parts is (6) 1 (5, 1) 1 1 , (4, 2) 1 1 1 (32) 1 0 1 17 and the characters of Young FG6-modules are given by Theorem 2.9 as follows. ch Y (6) = χ(6), ch Y (5,1) = χ(6) + χ(5,1), ch Y (4,2) = χ(5,1) + χ(4,2), 2 2 ch Y (3 ) = χ(6) + χ(5,1) + χ(4,2) + χ(3 ). Similar to the strategy in the proof of Example 2.10, we may compute the radical series of F (6) (5,1) (4,2) (32) Young G6-modules. Then, we can show that SB = EndFG6 (Y ⊕ Y ⊕ Y ⊕ Y )
is isomorphic to K4 := FQ/I with
α1 α2 α3 / / / α1β1,α2β2, β3α3,α1α2α3, β3β2β1, Q : ◦ o ◦ o ◦ o ◦ and I : . (4.1) β1 β2 β3 *β1α1α2 − α2α3β3, β2β1α1 − α3β3β2+
(See [DEMN, 3.5] for another method to show this.) Since β1α1 + α3β3 and β3β2α2α3 are non-trivial central elements of K4, the number of (pairwise non-isomorphic basic) support
τ-tilting modules for K4 is the same as K4/ hβ1α1,α3β3, β3β2α2α3i by Proposition 2.12. Then, similar to the proof method of Lemma 3.3, we have
s 012 3 4 #sτ-tilt K4 . as(K4) 1 4 12 36 83 136
This implies that SB is τ-tilting finite. Hence, the statement follows from the fact that
SB is the basic algebra of S(2, 6).
(2) The group algebra FG13 contains two blocks, i.e., the principal block B1 and the λ µ block B2 labeled by 2-core (2, 1). The parts of the decomposition matrix [S : D ] for the partitions in B1 and B2 with at most two parts are (13) 1 (12, 1) 1 (11, 2) 1 1 B1 : , B2 : (10, 3) 0 1 . (9, 4) 1 1 1 (8, 5) 1 0 1 (7, 6) 1011 We may prove that SB1 is isomorphic to K4, because the characters of Young modules for FG13 are as follows. (One may compare this with the case of S(2, 6)). ch Y (13) = χ(13), ch Y (11,2) = χ(13) + χ(11,2), ch Y (9,4) = χ(11,2) + χ(9,4), ch Y (7,6) = χ(13) + χ(11,2) + χ(9,4) + χ(7,6). F On the other hand, SB2 is isomorphic to A2 ⊕ by [Er, Proposition 4.1] as we have explained at the start of subsection 3.1. Therefore, the basic algebra of S(2, 13) is K4 ⊕
A2 ⊕ F, which is also τ-tilting finite.
18 (3) The group algebra FG15 also contains two blocks and the parts of the decomposi- tion matrix [Sλ : Dµ] for the partitions with at most two parts are as follows.
(15) 1 (14, 1) 1 (13, 2) 0 1 (12, 3) 1 1 , . (11, 4) 0 0 1 (10, 5) 1 1 1 (9, 6) 0101 (8, 7) 1011 After computing the characters of Young FG15-modules by Theorem 2.9, we deduce that
the basic algebra of S(2, 15) is isomorphic to K4 ⊕A2 ⊕ F ⊕ F.
Now, we look at the case S(2, 8). Let B be the principal block of FG8, the part of the decomposition matrix [Sλ : Dµ] for the partitions in B with at most two parts is
(8) 1 (7, 1) 1 1 (6, 2) 0 1 1 . (5, 3) 0111 (42) 0101 On the other hand, Theorem 2.9 implies that the characters of Young FG8-modules are
ch Y (8) = χ(8), ch Y (7,1) = χ(8) + χ(7,1), ch Y (6,2) = χ(8) + χ(7,1) + χ(6,2), ch Y (5,3) = χ(6,2) + χ(5,3), 2 2 ch Y (4 ) = χ(8) + χ(7,1) + χ(6,2) + χ(5,3) + χ(4 ).
It is obvious that Y (8) = D(8) and we may find others as follows (I am grateful to Prof. 2 Ariki for showing me the structure of Y (4 )).
D(6,2) (7,1) (8) D(8) D D D(7,1) ✟ ❏❏❏ ✟ (7,1) (7,1) (6,2) ✟✟ (6,2) ✟✟ (5,3) (5,3) Y = D , Y = ✟✟ D ✟✟ , Y = D , ✟ ❏❏❏ ✟ D(8) ✟ ✟ D(7,1) D(8) D(7,1) D(6,2) (7,1) (8) D ■ D ■■■ D(5,3) D(6,2)
2 Y (4 ) = D(7,1) D(7,1) .
(6,2) (5,3) D ■ D ■■■ D(8) D(7,1)
19 λ µ λ µ Note that the dimension of HomFG8 (Y ,Y ) between two Young modules Y ,Y is equal λ µ to the inner product (ch Y , ch Y ). By direct calculation, we conclude that SB = (8) (7,1) (6,2) (5,3) (42) F EndFG8 (Y ⊕ Y ⊕ Y ⊕ Y ⊕ Y ) is isomorphic to L5 := Q/I with
α1 α2 α3 / 2 / / Q : (5, 3) o (4 ) o (6, 2) o (7, 1) and β1 O β2 β3 β4 α4 (8) (4.2) α β ,α α , β α , β α , β α , β β , β α β ,α α α ,α β α , β β β , I : 1 1 1 4 3 3 2 2 4 4 4 1 4 2 2 1 2 3 2 2 4 3 2 1 . * β1α1α2 − α2α3β3, β2β1α1 − α3β3β2,α2β2β1α1 − β1α1α2β2 +
Here, we replace each vertex in the quiver of SB by the partition λ associated with the Young module Y λ. We have checked that there are at least 500 pairwise non-isomorphic
basic support τ-tilting L5-modules, so we leave this case to the future.
Proposition 4.2. Let p = 2. Then, the wild Schur algebras S(2, 17) and S(2, 19) are τ-tilting finite if and only if S(2, 8) is τ-tilting finite.
Proof. We show that the basic algebra of S(2, 17) is isomorphic to L5 ⊕A2 ⊕ F ⊕ F. The λ µ blocks of FG17 and the parts of the decomposition matrix [S : D ] for the partitions with at most two parts are as follows. (17) 1 (16, 1) 1 (15, 2) 1 1 (14, 3) 0 1 (13, 4) 0 1 1 , . (12, 5) 0 0 1 (11, 6) 0111 (10, 7) 0101 (9, 8) 01011 In order to identity L5, it suffices to check the characters of Young FG17-modules: ch Y (17) = χ(17), ch Y (15,2) = χ(17) + χ(15,2), ch Y (13,4) = χ(17) + χ(15,2) + χ(13,4), ch Y (11,6) = χ(13,4) + χ(11,6), ch Y (9,8) = χ(17) + χ(15,2) + χ(13,4) + χ(11,6) + χ(9,8).
For the case S(2, 19), the blocks of FG19 and the parts of the decomposition matrix [Sλ : Dµ] for the partitions with at most two parts are (19) 1 (18, 1) 1 (17, 2) 0 1 (16, 3) 1 1 (15, 4) 1 0 1 , (14, 5) 0 1 1 . (13, 6) 0001 (12, 7) 0111 (11, 8) 00101 (10, 9) 01011 Also, by Theorem 2.9, we have
20 ch Y (19) = χ(19), ch Y (15,4) = χ(19) + χ(15,4), ch Y (11,8) = χ(19) + χ(15,4) + χ(11,8); ch Y (18,1) = χ(18,1), ch Y (16,3) = χ(18,1) + χ(16,3), ch Y (14,5) = χ(18,1) + χ(16,3) + χ(14,5), ch Y (12,7) = χ(14,5) + χ(12,7), ch Y (10,9) = χ(18,1) + χ(16,3) + χ(14,5) + χ(12,7) + χ(10,9); ch Y (17,2) = χ(17,2); ch Y (13,6) = χ(13,6).
Similar to the above, the basic algebra of S(2, 19) is isomorphic to D3 ⊕L5 ⊕F⊕F, where
D3 is a τ-tilting finite algebra defined in subsection 3.2.
Proposition 4.3. Let p =2.
(1) If r is even, the Schur algebra S(2,r) is τ-tilting infinite for any r > 10. (2) If r is odd, the Schur algebra S(2,r) is τ-tilting infinite for any r > 21.
Proof. We denote by S(2,r) the basic algebra of S(2,r) and we use Lemma 2.16 to determine the quiver of S(2,r). When we display the quiver of S(2,r), we usually replace each vertex by the partition λ associated with the Young module Y λ. Then, the quiver of S(2, 10) is
/ (6, 4) o (10) , O O
/ / 2 / (8, 2) o (7, 3) o (5 ) o (9, 1) and the quiver of S(2, 21) is
/ (17, 4) (13, 8) o (21) (14, 7) (18, 3) . O O O / / / / (15, 6) o (11, 10) o (19, 2) (20, 1) o (12, 9) o (16, 5) Now, it is enough to say that S(2, 10) and S(2, 21) are τ-tilting infinite by Lemma 2.15. Hence, the statement follows from Lemma 2.14.
Proposition 4.4. Let p =2.
(1) The wild Schur algebra S(3, 5) is τ-tilting finite. (2) The Schur algebra S(n, r) is τ-tilting infinite for any n > 3 and r > 6.
Proof. We consider the Young modules Y λ for partitions λ with at most three parts. Then, Specht modules Sµ in the Specht filtration of Y λ and composition factors Dµ which appear in Y λ are also corresponding to the partitions with at most three parts. (1) We show that the basic algebra of S(3, 5) is τ-tilting finite. The group algebra
FG5 contains only two blocks, i.e., the principal block B1 and the block B2 labeled by λ µ 2-core (2, 1). The parts of the decomposition matrix [S : D ] for the partitions in B1
and B2 with at most three parts are as follows.
21 (5) 1 (3, 2) 1 1 B1 : , B2 : (4, 1) 1 . (3, 12) 2 1 (22, 1) 1 1 Combining with [Er, Proposition 5.8], the basic algebra of S(3, 5) is isomorphic to U4 ⊕F,
where U4 := FQ/I is presented by
α1 α2 α3 / / / Q : ◦ o ◦ o ◦ o ◦ and I : α1β1,α2β2,α1α2α3, β3β2β1,α3β3 − β2α2 . (4.3) β1 β2 β3 D E
Since β2α2 + β3α3 and β2β1α1α2 are non-trivial central elements of U4, the number of support τ-tilting modules of U4 is the same as U4/ hβ2α2, β3α3, β2β1α1α2i by Proposition 2.12. Then, by direct calculation, we have
s 012 3 4 #sτ-tilt U4 . as(U4) 1 4 12 36 83 136 We can also verify the number 136 by the String Applet [Ge]. (2) We shall show that S(3, 6), S(3, 7) and S(3, 8) are τ-tilting infinite. Then, the statement follows from Lemma 2.13 and Lemma 2.14. As we are already familiar with the strategy of determining the radical series of Young modules and the basic algebras of Schur algebras, we may leave this heavy work to a computer and some mathematicians indeed did. Here, we refer to Carlson and Matthews’s program [CM].
(2.1) Let B be the principal block of FG6, the part of the decomposition matrix [Sλ : Dµ] for the partitions in B with at most three parts is of the form
(6) 1 (5, 1) 1 1 (4, 2) 1 1 1 . (4, 12) 2 1 1 (32) 1 0 1 (23) 1 0 1 λ Then, the quiver of SB = EndFG6 ( Y ) is as follows. λ∈B∩Ω(3,6) L ◦ / ◦ / ◦ O o O o O
/ / ◦ o ◦ o ◦
(2.2) Let B be the principal block of FG7, the part of the decomposition matrix [Sλ : Dµ] for the partitions in B with at most three parts is of the form
22 (7) 1 (5, 2) 0 1 (4, 2, 1) 1 1 1 . (5, 12) 1 1 0 (32, 1) 1 0 1 (3, 22) 1 0 1 λ Then, the quiver of SB = EndFG7 ( Y ) is as follows. λ∈B∩Ω(3,7) L / / ◦ o ◦ o ● ◦ O O c●●●● ●●●● ●●●● ●●●● / ● # ◦ o ◦ ◦
(2.3) Let B be the principal block of FG8, the part of the decomposition matrix [Sλ : Dµ] for the partitions in B with at most three parts is of the form
(8) 1 (7, 1) 1 1 (6, 2) 0 1 1 (5, 3) 0111 (4, 3, 1) 21111. (6, 12) 11100 (42) 01010 (4, 22) 20101 2 (3 , 2) 20001 λ Then, the quiver of SB = EndFG8 ( Y ) is as follows. λ∈B∩Ω(3,8) L ◦ / ◦ / ◦ O o O o O ◦ / ◦ ◦ o O ②< ②②②② ②②②② / ②|②② / ◦ o ◦ o ◦ By Lemma 2.15, we conclude that S(3, 6), S(3, 7) and S(3, 8) are τ-tilting infinite.
Corollary 4.5. Let p =2. The wild Schur algebra S(4, 5) is τ-tilting finite.
Proof. We consider the Young modules Y λ for partitions λ of 5 with at most four parts. Note that S(3, 5) is an idempotent truncation of S(4, 5) as we mentioned in Lemma 2.13. Compared with the case S(3, 5), the case S(4, 5) has only one additional partition (2, 13)
which appears in the block of FG5 labeled by 2-core (2,1). Then, the basic algebra of
S(4, 5) is isomorphic to U4 ⊕A2 based on the result of S(3, 5).
23 Table 2: The τ-tilting finite S(n, r) over p = 3. r 1 2 3 4 5 6 7 8 9 10 11 12 13 ··· n 2 S S F F F F F F T T T W W ··· 3 S S F F F W T T W W W W W ··· 4 S S F F F W W W W W W W W ··· 5 S S F F F W W W W W W W W ··· ......
Proposition 4.6. Let p =2. The Schur algebra S(n, 4) is τ-tilting infinite for any n > 4. Proof. By our strategy in subsection 2.2, one can see that S(n, 4) with n > 5 is always Morita equivalent to S(4, 4). So, it is enough to show that S(4, 4) is τ-tilting infinite. In fact, the quiver of the basic algebra of S(4, 4) displayed below implies our statement. ◦ / ◦ O o O
/ / ◦ o ◦ o ◦ This quiver has been given by Xi in [Xi].
Now, we have already determined the τ-tilting finiteness of wild Schur algebras over p = 2, except for S(2,r) with r =8, 17, 19, S(3, 4) and S(n, 5) with n > 5. Remark 4.7. Let p = 2. The Schur algebra S(n, 5) with n > 6 is always Morita
equivalent to S(5, 5). Moreover, the basic algebra of S(5, 5) is isomorphic to N5 ⊕A2
following [Xi, Proposition 3.8], where N5 := FQ/I is presented by
α1 α2 α3 α4 / / / / Q : ◦ o ◦ o ◦ o ◦ o ◦ with β1 β2 β3 β4 (4.4) α β ,α β ,α β , β α ,α α α α , β β β β , β α − α α β β , I : 1 1 2 2 3 3 4 4 1 2 3 4 4 3 2 1 2 2 3 4 4 3 . * α2α3α4β4 − β1α1α2α3, β3β2β1α1 − α4β4β3β2 +
4.2 The characteristic p =3
We assume in this subsection that the characteristic of F is 3. Then, the τ-tilting finiteness for S(n, r) is shown in Table 2 and the proof is divided into the propositions displayed below. Here, we use the same conventions with Table 1. Proposition 4.8. Let p =3. The Schur algebra S(2,r) is τ-tilting infinite for r > 12. Proof. We show that both S(2, 12) and S(2, 13) are τ-tilting infinite. Then, the statement
follows from Lemma 2.14. In fact, let B be the principal block of FG12 and the quiver of λ SB = EndFG12 ( Y ) following Lemma 2.16 is λ∈B∩Ω(2,12) L 24 / (11, 1) o (12) , O O
/ / 2 (9, 3) o (8, 4) o (6 ) where we replace each vertex by the partition λ associated with Y λ. Thus, S(2, 12) is τ-tilting infinite by Lemma 2.15. One can check that S(2, 13) also contains a τ-tilting infinite subquiver as shown above.
In the following, we refer to [CM] for the quiver of SB without further notice.
Proposition 4.9. Let p =3. Then, S(3, 6) and S(3,r) with r > 9 are τ-tilting infinite.
Proof. It suffices to show that S(3, 6), S(3, 10) and S(3, 11) are τ-tilting infinite. λ µ (1) Let B be the principal block of FG6, the part of the decomposition matrix [S : D ] for the partitions in B with at most three parts is of the form (6) 1 (5, 1) 1 1 (4, 12) 0 1 1 . 2 (3 ) 0101 (3, 2, 1) 11111 (23) 10001 λ Then, the quiver of SB = EndFG6 ( Y ) is as follows. λ∈B∩Ω(3,6) L 9 ◦ ❑ sss e❑❑❑❑ ssss ❑❑❑❑ ssss ❑❑❑❑ / syss / ❑/ % ◦ o ◦ o ❑❑ ◦ o s9 ◦ e❑❑ ❑❑ ss s ❑❑ ❑❑ ss ss ❑❑ ❑ ss ss ❑ % ◦ yss
(2) Let B1 be the principal block of FG10 and B2 the block of FG11 labeled by 3-core 2 λ µ (1 ), the parts of the decomposition matrix [S : D ] for the partitions in B1 and B2 with at most three parts are of the form (10) 1 (10, 1) 1 (8, 2) 1 1 (9, 2) 1 1 (7, 3) 0 1 1 (7, 4) 0 1 1 B : , B : . 1 2 2 (7, 2, 1) 1111 (7, 2 ) 1111 2 (5 ) 00101 (6, 5) 00101 (4, 32) 001111 (42, 3) 001111 Then, both the quiver of SB1 and SB2 are as follows. ◦ / ◦ / ◦ o O o
/ / ◦ o ◦ o ◦ By Lemma 2.15, the above two cases are τ-tilting infinite quivers.
25 Proposition 4.10. Let p =3. Then, S(n, r) is τ-tilting infinite for any n > 4 and r > 6.
Proof. Based on the result of S(3,r), Lemma 2.13 and Lemma 2.14, it suffices to show that S(4, 7) and S(4, 8) are τ-tilting infinite. λ µ (1) Let B be the principal block of FG7, the part of the decomposition matrix [S : D ] for the partitions in B with at most four parts is of the form (7) 1 (5, 2) 1 1 (4, 3) 0 1 1 (4, 2, 1) 1111 . (3, 2, 12) 10111 3 (4, 1 ) 00010 3 (2 , 1) 10001 λ Then, the quiver of SB = EndFG7 ( Y ) is as follows. λ∈B∩Ω(4,7) L ◦ 5/ ◦ O o ❧❧❧❧ O ❧❧❧❧❧❧ ❧❧❧❧❧❧ ❧❧❧❧❧❧ ◦ / ◦ ❘u❧❘❧ / ◦ O o i❘o ❘❘❘❘ ❘❘❘❘❘❘ ❘❘❘❘❘❘ ❘❘❘❘/) ◦ o ❘ ◦ 2 (2) Let B be the block of FG8 labeled by 3-core (1 ), the part of the decomposition matrix [Sλ : Dµ] for the partitions in B with at most four parts is of the form (7, 1) 1 (6, 2) 1 1 (42) 0 1 1 . 2 (4, 2 ) 1111 (3, 22, 1) 10011 λ Then, the quiver of SB = EndFG8 ( Y ) is as follows. λ∈B∩Ω(4,8) L ◦ / ◦ O o O
/ / ◦ o ◦ o ◦ Obviously, S(4, 7) and S(4, 8) are τ-tilting infinite.
Hence, we have determined the τ-tilting finiteness for all the cases over p = 3.
4.3 The characteristic p > 5
The situation on p > 5 is much easier than the situation on p = 2, 3. As shown in Proposition 3.1, tame Schur algebras do not appear in this case. Then, the τ-tilting finiteness for S(n, r) is shown in Table 3 and the proof is divided into two propositions. Also, we use the same conventions with Table 1.
26 Table 3: The τ-tilting finite S(n, r) over p > 5. r 1 ∼ p − 1 p ∼ 2p − 1 2p ∼ p2 − 1 p2 ∼ p2 + p − 1 p2 + p ∼∞ n 2 S F F W W 3 S F W W W 4 S F W W W 5 S F W W W ......
Proposition 4.11. Let p > 5. Then, S(2,r) is τ-tilting infinite for any r > p2 + p.
Proof. It suffices to consider S(2,p2 + p) and S(2,p2 + p + 1) following Lemma 2.14. To show the τ-tilting finiteness of S(2,p2 + p), we choose four partitions (p2 + p), (p2 + p − 1, 1), (p2 − p, 2p), (p2 − 1,p + 1),
which are contained in the principal block B of FGp2+p. By Lemma 2.16, one may
construct the following subquiver of the quiver of SB.
2 2 v(p +p) / v(p +p−1,1) O o O
(p2−p,2p) / (p2−1,p+1) v o v 2 This is just the τ-tilting infinite quiver Q1 and therefore, S(2,p + p) is τ-tilting infinite. 2 Moreover, we can show that S(2,p +p+1) contains the τ-tilting infinite quiver Q1 as a subquiver if we choose (p2 + p +1), (p2 + p − 1, 2), (p2 − p +1, 2p) and (p2 − 1,p + 2).
Proposition 4.12. Let p > 5. Then, the Schur algebra S(n, r) is τ-tilting infinite for any n > 3 and r > 2p.
Proof. It suffices to consider S(3,r) for r =2p+x with 0 6 x 6 2. Let B be the principal λ µ block of FGr. Then, the part of the decomposition matrix [S : D ] for the partitions in B with at most three parts is of the form (2p + x) 1 (2p − 1, 1+ x) 1 1 (p + x, p) 0 1 1 . (2p − 2, 1+ x, 1) 0101 (p + x, p − 1, 1) 11111 ((p − 1)2, 2+ x) 100011 We recall from [Er, Proposition 5.3.1] that the quiver of SB is
27 9 ◦ ❑ sss e❑❑❑❑ ssss ❑❑❑❑ ssss ❑❑❑❑ / syss / ❑/ % ◦ o ◦ o ❑❑ ◦ o s9 ◦ . e❑❑ ❑❑ ss s ❑❑ ❑❑ ss ss ❑❑ ❑ ss ss ❑ % ◦ yss Then, the statement follows from Lemma 2.13, Lemma 2.14 and Lemma 2.15.
Hence, we have already determined the τ-tilting finiteness of wild Schur algebras over p > 5, except for S(2,r) with p2 6 r 6 p2 + p − 1.
4.4 The remaining cases
So far, we have determined the τ-tilting finiteness of Schur algebras, except for the cases in (⋆). As we explained in Proposition 4.2 and Remark 4.7, when p = 2, the cases S(2,r) for r = 17, 18 actually reduce to the principal block of S(2, 8) and the cases S(n, 5) for n > 6 can be reduced to the principal block of S(5, 5). Actually, the cases S(2,r) for p2 6 r 6 p2 + p − 1 over p > 5 also reduce to a block case if one applies [EH2, Theorem 13], which says that any two blocks of S(2,r) with the same number of simples are Morita equivalent over the same field. We will deal with them separately in the future. Now, we may look at S(3, 4) over p = 2. We recall from [DEMN, 3.6] that the basic
algebra of S(3, 4) is presented by the bound quiver algebra M4 := FQ/I with
α1 α2 α β , β α ,α α , β β , Q : ◦ / ◦ / ◦ and I : 1 1 3 3 1 2 2 1 . (4.5) o O o β1 β2 α1α3β3,α3β3β1, β1α1 − α2β2 β3 α3 * + ◦
We have the following partial results. (Recall that as(A) is the number of pairwise non- isomorphic basic support τ-tilting A-modules M with |M| = s for 0 6 s 6 |A|.)
s 01 2 3 . as(M4) 1 4 12 40
On the other hand, let P4 be the preprojective algebra of type D4 which has been studied by Mizuno [Mi]. We recall his result as follows.
s 012 3 4 #sτ-tilt P4 . as(P4) 1 4 12 40 135 192
Then, we expect that a4(M4) 6 135 according to our experiences in the calculation process. More generally, we expect that the following conjecture is true. A Let |A| = n and Pi the indecomposable projective A-modules. The Cartan matrix [cij] A F for i, j ∈ {1, 2,...,n} is defined by cij = dim HomA(Pi,Pj). Suppose that A := Q/I1 and B := FQ/I2 are two algebras given by the same quiver Q with different admissible ideals I1 and I2. It is obvious that a0(A)= a0(B) = 1 and a1(A)= a1(B)= |Q0|.
28 Conjecture 4.13. Assume that A and B have the same quiver and B is τ-tilting finite. A B If cij 6 cij for any i, j ∈{1, 2,..., |Q0|}, then as(A) 6 as(B) for 2 6 s 6 |Q0|.
For example, let Q be the following quiver.
1 / 2 / 3 o O o 4
We consider the preprojective algebra P4 and the tame block algebra D4 (or H4) defined in subsection 3.2. Then, we have
1101 2211 D4 1312 P4 2422 [cij ]= and [cij ]= . 0120 1221 1202 1212 D4 P4 It is easy to check that cij 6 cij for any i, j ∈{1, 2, 3, 4}. Moreover, we have shown that a2(D4)=12= a2(P4), a3(D4)=36 < 40 = a3(P4) and a4(D4)=61 < 135 = a4(P4). Besides, Conjecture 4.13 is proved to be true by [W1, Theorem 1.1] for the two-point algebras presented by the following quivers with some relations.
◦ / ◦ e , ◦o ◦ e . Finally, we have the following conjecture on Schur algebras.
Conjecture 4.14. All wild Schur algebras contained in (⋆) are τ-tilting finite.
A A complete list of τ-tilting D4-modules in the proof of Lemma 3.3. f We recall the indecomposable projective D4-modules Pi as follows.
2 3 1 3 1 2 4 4 P1 = 3 ,P2 = 1 ,P3f= 3 , P4 = 3 . 2 3 2
Then, we construct three indecomposable D4-modules to describe some basic τ-tilting modules. We first consider the τ-tilting D4-module P1 ⊕ P2 ⊕ P3 ⊕ P4 and take an exact f sequence with a minimal left add(P1 ⊕ P2 ⊕ P4)-approximation π1 of P3: f π1 P3 −→ P1 ⊕ P2 ⊕ P4 −→ coker π1 −→ 0.
We define M1 := coker π1 and P1 ⊕ P2 ⊕ M1 ⊕ P4 is again a τ-tilting D4-module. Then, we take an exact sequence with a minimal left add(P2 ⊕ M1 ⊕ P4)-approximation π2: f π2 P1 −→ P2 ⊕ M1 −→ coker π2 −→ 0
29 3 3 3 1 1 4 1 1 2 and define M2 := coker π2. Last, we consider the τ-tilting D4-module 3 ⊕ 3 ⊕ 3 ⊕ 3 (one 2 2 2 2 may check this by the definition) and define M3 := coker π3 as the cokernel of π3, where 3 3 3 f 1 4 1 1 2 π3 is a minimal left add(3 ⊕ 3 ⊕ 3 )-approximation with the following exact sequence. 2 2 2 3 3 1 π3 1 4 1 2 3 −→ 3 ⊕ 3 −→ coker π3 −→ 0. 2 2 2
Using P1,P2,P3,P4, M1, M2, M3 and other explicitly described modules, we can give a complete list of τ-tilting D4-modules by direct computation of left mutations.
3 3f 1 4 P1 ⊕ P2 ⊕ P3 ⊕ P4 2 4 ⊕ P2 ⊕ P3 ⊕ P4 P1 ⊕ 3 ⊕ P3 ⊕ P4 P1 ⊕ P2 ⊕ M1 ⊕ P4 2
3 3 3 1 2 3 1 4 3 2 3 1 2 P1 ⊕ P2 ⊕ P3 ⊕ 3 2 4 ⊕ 3 ⊕ P3 ⊕ P4 2 4 ⊕ P2 ⊕ 3 ⊕ P4 2 4 ⊕ P2 ⊕ P3 ⊕ 3 2 2 2
3 1 4 1 1 2 1 4 P1 ⊕ 3 ⊕ 3 ⊕ P4 P1 ⊕ P2 ⊕ M1 ⊕ P1 ⊕ ⊕ M1 ⊕ P4 M2 ⊕ P2 ⊕ M1 ⊕ P4 2 3 3
3 3 3 1 4 1 2 3 1 4 3 2 3 2 3 P1 ⊕ 3 ⊕ P3 ⊕ 3 2 4 ⊕ 3 ⊕ 4 ⊕ P4 M2 ⊕ P2 ⊕ 3 ⊕ P4 2 4 ⊕ P2 ⊕ 3 ⊕ 2 2 2 2
3 3 3 3 3 3 3 1 2 1 4 1 4 1 1 4 1 1 4 1 1 2 4 ⊕ P2 ⊕ 2 ⊕ 3 3 ⊕ 3 ⊕ 3 ⊕ P4 P1 ⊕ 3 ⊕ 3 ⊕ P4 P1 ⊕ 3 ⊕ 3 ⊕ 3 2 2 2 2
1 2 1 4 1 2 2 4 1 4 2 4 M2 ⊕ P2 ⊕ M1 ⊕ 3 P1 ⊕ 3 ⊕ M1 ⊕ 3 3 ⊕ 3 ⊕ M1 ⊕ P4 M2 ⊕ 3 ⊕ M1 ⊕ P4
3 3 3 3 3 3 3 3 1 4 1 1 2 1 4 1 2 1 4 1 4 3 2 4 2 P1 ⊕ 3 ⊕ 3 ⊕ 3 2 4 ⊕ 3 ⊕ P3 ⊕ 3 3 ⊕ 3 ⊕ 4 ⊕ P4 M2 ⊕ 3 ⊕ 3 ⊕ P4 2 2 2 2 2 2
2 3 3 3 2 3 1 4 1 4 1 1 1 4 1 2 4 1 2 M2 ⊕ P2 ⊕ 3 ⊕ 1 2 3 ⊕ 3 ⊕ 3 ⊕ 3 P1 ⊕ 3 ⊕ 3 ⊕ S1 M2 ⊕ 3 ⊕ M1 ⊕ 3 3 2 2 2 3 1 2 2 4 1 4 1 2 1 4 1 2 2 4 1 4 M2 ⊕ P2 ⊕ 1 2 ⊕ ⊕ ⊕ M1 ⊕ P1 ⊕ ⊕ S1 ⊕ ⊕ ⊕ S4 ⊕ P4 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 2 1 4 1 1 2 1 4 1 2 1 4 1 4 3 1 2 4 2 3 M3 ⊕ 3 ⊕ 3 ⊕ 3 2 4 ⊕ 3 ⊕ M3 ⊕ 3 3 ⊕ 3 ⊕ 4 ⊕ 3 M2 ⊕ 3 ⊕ 3 ⊕ 1 2 2 2 2 2 2 2 2 3
3 3 3 2 1 4 1 1 1 2 4 3 1 2 2 4 1 4 124 1 2 124 1 4 1 2 3 ⊕ ⊕ 3 ⊕ 3 M2 ⊕ 3 ⊕ 1 2 ⊕ 3 3 ⊕ 3 ⊕ 3 ⊕ 3 3 ⊕ 3 ⊕ S1 ⊕ 3 3 2 3 3 3 3 3 3 2 4 1 4 124 3 1 1 2 1 4 1 3 3 1 4 3 3 ⊕ 3 ⊕ S4 ⊕ 3 M3 ⊕ 2 ⊕ 3 ⊕ 3 M3 ⊕ 3 ⊕ 3 ⊕ 4 2 4 ⊕ 3 ⊕ M3 ⊕ 4 2 2 2 2 2 3 3 2 2 3 1 2 3 3 3 1 4 1 3 1 2 4 2 3 2 4 3 1 2 2 4 ⊕ 2 ⊕ M3 ⊕ 3 3 ⊕ ⊕ 4 ⊕ 3 S2 ⊕ 3 ⊕ 3 ⊕ 1 2 S2 ⊕ 3 ⊕ 1 2 ⊕ 3 2 3 2 3 3
2 4 124 1 2 124 1 2 124 1 4 2 4 124 3 ⊕ S2 ⊕ 3 ⊕ 3 3 ⊕ S2 ⊕ S1 ⊕ 3 3 ⊕ 3 ⊕ S1 ⊕ S4 3 ⊕ S2 ⊕ S4 ⊕ 3
3 3 3 1 3 3 3 3 3 3 1 124 M3 ⊕ 2 ⊕ 3 ⊕ 4 2 4 ⊕ 2 ⊕ M3 ⊕ 4 S3 ⊕ 1 ⊕ 4 ⊕ 3 3 ⊕ S2 ⊕ S1 ⊕ S4 2 3 2 3 3 1 3 S3 ⊕ 2 ⊕ 3 ⊕ 4 2
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[Ada] T. Adachi, Characterizing τ-tilting finite algebras with radical square zero. Proc. Amer. Math. Soc. 144 (2016), no. 11, 4673–4685. [Asa] S. Asai, Semibricks. Int. Math. Res. Not. IMRN (2020), no.16,4993–5054. [Ao] T. Aoki, Classifying torsion classes for algebras with radical square zero via sign decomposition. Preprint (2018), arXiv: 1803.03795. [AAC] T. Adachi, T. Aihara and A. Chan, Classification of two-term tilting com- plexes over Brauer graph algebras. Math. Z. 290 (2018), no. 1–2, 1–36. [AHMW] T. Aihara, T. Honma, K. Miyamato and Q. Wang, Report on the finiteness of silting objects. To appear in Proceedings of the Edinburgh Mathematical Society. [AIR] T. Adachi, O. Iyama and I. Reiten, τ-tilting theory. Compos. Math. 150 (2014), no. 3, 415–452. [AIRRT] L. Demonet, O. Iyama, N. Reading, I. Reiten and H. Thomas, Lattice theory of torsion classes. Preprint (2017), arXiv: 1711.01785. [ASS] I. Assem, D. Simson and A. Skowronski´ , Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory. London Mathematical Society Student Texts, 65. Cambridge University Press, Cambridge, 2006. [CM] J. F. Carlson, and G. Matthews, Schur Algebras, S(n, r). http://alpha.math.uga.edu/ jfc/schur.html. [DEMN] S.R. Doty, K. Erdmann, S. Martin and D.K. Nakano, Representation type of Schur algebras. Math. Z. 232 (1999), no. 1, 137–182. [DIJ] L. Demonet, O. Iyama and G. Jasso, τ-tilting finite algebras, bricks and g-vectors. Int. Math. Res. Not. (2017), no. 00, pp. 1–41. [DN] S.R. Doty and D.K. Nakano, Semisimple Schur algebras. Math. Proc. Cam- bridge Philos. Soc. 124 (1998), no. 1, 15–20. [DR] S. Donkin and I. Reiten, On Schur algebras and related algebras V: Some quasi- hereditary algebras of finite type. J. Pure Appl. Algebra 97 (1994), no. 2, 117–134. [Er] K. Erdmann, Schur algebras of finite type. Quart. J. Math. Oxford Ser. 44 (1993), no. 173, 17–41. [EH1] K. Erdmann and A. Henke, On Schur algebras, Ringel duality and symmetric groups. J. Pure Appl. Algebra 169 (2002), no. 2-3, 175–199. [EH2] K. Erdmann and A. Henke, On Ringel duality for Schur algebras. Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 1, 97–116. [EJR] F. Eisele, G. Janssens and T. Raedschelders, A reduction theorem for τ-rigid modules. Math. Z. 290 (2018), no. 3-4, 1377–1413. [Ge] J. Geuenich, String Applet: https://www.math.uni-bielefeld.de/ jgeuenich/string-applet/. [He] A. Henke, Young modules and Schur subalgebras. Ph.D. thesis Linacre College,
31 Oxford, 1999. [Ja] G.D. James, The representation theory of the symmetric groups. Lecture Notes in Mathematics, 682. Springer, Berlin, 1978. [Ma] S. Martin, Schur algebras and representation theory. Cambridge Tracts in Math- ematics, 112. Cambridge University Press, Cambridge, 1993. [Mi] Y. Mizuno, Classifying τ-tilting modules over preprojective algebras of Dynkin type. Math. Z. 277 (2014), no. 3-4, 665–690. [Mo] K. Mousavand, τ-tilting finiteness of biserial algebras. Preprint (2019), arXiv: 1904.11514. [MS] P. Malicki and A. Skowronski´ , Cycle-finite algebras with finitely many τ-rigid indecomposable modules. Comm. Algebra 44 (2016), no. 5, 2048–2057. [Pl] P. G. Plamondon, τ-tilting finite gentle algebras are representation-finite. Pacific J. Math. 302 (2019), no. 2, 709–716. [Sa] B.E. Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Graduate Texts in Mathematics, 203. Springer- Verlag, New York, 2001. [Ve] M. Vernon, On a construction of Young modules. Ph.D. thesis, University of Leicester, 2005. [W1] Q. Wang, τ-tilting finiteness of two-point algebras I. Preprint (2019), arXiv: 1902.03737. [W2] Q. Wang, On τ-tilting finite simply connected algebras. Preprint (2019), arXiv: 1910.01937. [Xi] C. Xi, On representations types of q-Schur algebras. J. Pure Appl. Algebra 84 (1993), no. 1, 73–84. [Zi] S. Zito, τ-tilting finite cluster-tilted algebras. Proc. Edinb. Math. Soc. (2) 63 (2020), no. 4, 950–955.
Department of Pure and Applied Mathematics, Graduate School of Information Sci- ence and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka, 565-0871, Japan. Email address: [email protected]
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