Representation type, boxes, and Schur algebras

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2 Notation k algebraically closed field char k = p 0 A finite dimensional k-algebra mod A category of finite dimensional (left) A-modules M mod A [M], the isomorphism class of M ind A = {[M] M mod A indecomposable} ind d A = {[M] M ind A, dim M = d}

3 Finite representation type Definition An algebra A is called representation-finite if ind A <. Otherwise it is called representation-infinite. Examples semisimple algebras are representation-finite kc n is representation-finite k(c 2 C 2 ) is representation-infinite for char k = 2

4 Schur algebras Let V = k n. GL n V d Σ d The Schur algebra occurs in two ways: S(n, d) = End kσd (V d ) = Im(k GL n End(V d )) Theorem (Schur 1901) There is an idempotent e S(n, d), such that mod S(n, d) mod es(n, d)e = mod kσ d M em is an equivalence for char k = 0 and n d. For n d: mod S(n, d) = category of strict polynomial functors

5 Representation-finite blocks of Schur algebras Theorem (Xi 92 93, Erdmann 93, Donkin, Reiten 94) The Schur algebra S(n, d) has finite representation type exactly in the following cases: 1 n 3, d < 2p 2 n = 2, d < p 2 3 n = 2, d = 5, 7 The representation-finite blocks of S(n, d) are Morita equivalent to the path algebra of α m β 1 α 2 β 2 α m 1 β m 1 with relations α i 1 β i 1 = β i α i, α m 1 β m 1, α i α i 1, β i 1 β i

6 Tame and wild for A representation-infinite Definition An algebra A is called tame if for each dimension d there exist finitely many A-k[x]-bimodules N (d) 1,..., N (d) m(d), finitely generated as k[x]-modules, such that { } ind d A [N (d) i X ] X mod k[x], i = 1,..., m(d). Definition An algebra A is called wild if there is an A-k x, y -bimodule N such that N is finitely generated projective as a k x, y -module and N k x,y : mod k x, y mod A preserves indecomposability and isomorphism classes.

7 Tame-wild dichotomy Theorem (Drozd 80) Any finite dimensional algebra is either representation-finite, tame or wild.

8 Tame Schur algebras Theorem (Doty, Erdmann, Martin, Nakano 99) The Schur algebra S(n, d) is tame exactly in the following cases: 1 p = 2, n = 2, d = 4, 9 2 p = 3, n = 3, d = 7 3 p = 3, n = 3, d = 8 4 p = 3, n = 2, d = 9, 10, 11

9 Blocks of tame Schur algebras Theorem (Continued) Their blocks are Morita equivalent to the following quivers with relations: α , β 1 α 1 = β 2 α 2 = α 2 β 2 α 1 = β 1 α 2 β 2 = 0 α 2 β 1 α 3 β β 3 α 1 β 2 3 α 2 β 2 α 2 β β 1 α 3 α 1, β 1 β 3 α 3 = α 2 α 3 = β 3 β 2 = 0, α 3 β 3 = β 2 α 2, α 2 β 2 = β 1 α 1 β 1 α 1 = β 1 α 2 = β 1 α 3 = 0, β 2 α 1 = β 2 α 2 = β 3 α 1 = 0, α 3 β 3 = α 1 β 1 + α 2 β 2 β 1 α 1 = β 2 α 2 = β 1 α 3 = 0 β 2 α 3 = β 3 α 1 = β 3 α 2 = 0, α 2 β 2 = α 3 β 3

10 Representation type of subcategories Let A be a finite dimensional algebra. Question 1 Under which conditions does there exist a tame-wild dichotomy theorem for C mod A? 2 Is a particular C representation-finite, tame or wild?

11 Quasi-hereditary algebras Definition An algebra is called quasi-hereditary if there exist modules (i) with End( (i)) = k and Ext s ( (i), (j)) 0 i < j and A F( ), where F( ) := {N 0 = N 0 N 1 N t = N, N i /N i 1 = (ji )}. Example The Schur algebra S(n, d) is quasi-hereditary. The (i) are the Weyl modules. Theorem (Hemmer, Nakano 04) For char k = p > 3 and n d the Schur functor restricts to an equivalence F( ) F(S), where S are the Specht modules.

12 Filtered representation type Question 1 Does there exist a tame-wild dichotomy theorem for F( ) mod A? 2 For a particular class of quasi-hereditary algebras, is F( ) representation-finite, tame, or wild?

13 Differential biquivers - Motivation ( ) k 0 Let A = k k a = k( 1 2 ). Representations of A: V a V 1 V 2 ω 1 ω 2 W a W 1 W 2 0 k k id k id id Indecomposable representations 0 k, k k The following also describes mod A:, k 0 V 2 V 3 V 1 k 0 0 ω 2 ι ω 3 π ω 1 id W 2 W 3 W 1 0 k 0

14 Differential biquivers Definition 1 A biquiver is a quiver with two types of arrows, solid (deg = 0) and dashed (deg = 1). 2 A differential biquiver is a biquiver together with a linear map : kq kq of degree 1 such that (e i ) = 0 for all i and is a differential, i.e. 2 = 0 and (ab) = (a)b + ( 1) deg a a (b). Example b 4 b π ι (b) = b ι

15 Representations of differential biquivers Definition The category of representations of a differential biquiver is given as follows: Objects: Representations of the solid part. Morphisms: Pairs ((α i ) i Q0, (α ϕ ) ϕ dashed ) such that commutativity deformed by (solid) holds. Composition: Given by (dashed arrows) Example V b V b V 2 V 3 V 4 V 1 ω 2 ι ω 3 π ω 4 ω 1 W 2 W 3 W 4 W 1 W b W b ω 3 V b = W b ω 4 ω 3 V b = W b ω 2 + W b ι

16 Example of the reduction algorithm b a 5 4 b ι π b 5 (b) = b ι 5 ι ι π π b ι 4 π b (b ) = ι, (π ) = π π ι π ι π ι 4 π π b (π ) = π π (π ) = π π, ( ι) = ι ˆι ( π ) = π ι + ιˆπ ι ˆι ˆπ ι π π ι π π

17 Passing to a hereditary situation Definition Let P(A) be the category with objects P f Q, where P, Q proj A morphisms P α f f Q P Q β with βf = f α Let P 1 (A) be the full subcategory with Im f rad Q. Proposition The functor Coker: P 1 (A) \ (P 0) mod A, f Coker f is full, dense, preserves indecomposability and isomorphism classes.

18 Transforming P(A) to a differential biquiver Define the following biquiver: 2n vertices i and i Solid arrow i j for each basis vector of rad A (P i, P j ) Dashed arrow i j for each basis vector of rad A (P i, P j ) In P 1 (A): objects: P Q = morphisms: P n i i P α A ij P m j j f f Q P Q β In rep(q, ): objects: (k n i morphisms: k n i k n ĩ α iĩ on i) A ij (k m j A ij A ĩ j k m j k m j β j j on j),

19 Example of the reduction algorithm k[x]/(x 2 ) = kc 2 ϕ x 1 1 ϕ ϕ ϕ 22 ϕ22 2 ϕ 12 ϕ 21 ϕ 21 π ι ϕ ϕ (ϕ) = ιϕ 21 (ϕ) = ϕ 21 ι (ϕ 22 ) = ϕ 21 π (ϕ 12 ) = ϕπ + πϕ 22 (ϕ 22 ) = ιϕ 21 (ϕ 12 ) = ιϕ ϕ 22 ι 2 ι ϕ 21 1 ϕ

20 Directed biquivers Theorem (Koenig-K-Ovsienko 14) For every quasi-hereditary algebra A, there exists a directed biquiver (with relations) (Q, I, ) with F( ) = rep(q, I, ). No relations if and only if A is strongly quasi-hereditary, i.e. projdim (i) 1 for all i. Corollary For every strongly quasi-hereditary algebra A, F( ) is either representation-finite, tame or wild.

21 A strongly quasi-hereditary example A = k( 1 a 2 b )/(ab). (1) = S(1) = 1, (2) = P(2) = 2 1 Differential biquiver: (1) (2) Reduction: ϕ 33 a ϕ ϕ 31 π 1 (1) (2) = 2 1 ι ϕ 13 ϕ 13 = ϕπ ϕ 31 = ιϕ ϕ 33 = ιϕ 13 + ϕ 31 π 1 = (1) (2) = 2 ϕ 1

22 Boxes of tame Schur algebras Proposition (K 14) The following are the differential biquivers with relations for the tame Schur algebras: a ϕ r c χ c b ψ r d, (c) = bϕ, (r) = ba (χ) = ψϕ, (r) = eb a b e , (c) = bϕ + ψa, ϕ ψ ρ (d) = eψ + ρb

23 Boxes of tame Schur algebras Proposition (Continued) χ c t f σ (χ) = ψϕ, (t) = ea, a b , (c) = bϕ + ψa ϕ ψ (f ) = eϕ + σa e 4 t (r) = ba, (s) = db r s (t) = ea + dc, a b d, (c) = bϕ + ψa ϕ ψ χ (e) = dψ c e

24 Tame Schur algebras are filtered-finite Theorem (K-Thiel 14) The category F( ) is representation-finite for the tame Schur algebras. Proposition (Burt-Butler 91) If the underlying solid quiver with relations is of finite representation type, then F( ) is of finite representation type. Proof. Compute their differential biquivers with relations, for 1 & 4 use the Proposition, the underlying algebras are special biserial for 2 & 3 use a computer to reduce.

25 Filtered-finite wild Schur algebras Theorem (Erdmann, de la Peña, Sáenz 02) There are wild blocks of the Schur algebras S(2, p 2 ), S(2, p 2 + 1) with F( ) representation-finite, e.g. the following for n 5: m α 1 β 1 α 2 γ 3 m 2 m δ 3 γ 4 δ 4 γ n 1 δ n 1 m 1 β 2 with relations: β 1 α 1 = β 2 α 2 = β 2 δ 3 = β 1 δ 3 = γ 3 α 2 = γ 3 α 1 = 0 α 2 β 2 α 1 = β 1 α 2 β 2 = γ i+1 γ i = δ i δ i+1 = 0, δ 3 γ 3 = α 2 β 2, γ i δ i = δ i+1 γ i+1.

26 General strategy for filtered representation type Take your favourite quasi-hereditary algebra, determine the standard modules, compute the (A -structure on the) Ext-algebra of the standard modules, dualise, to get a differential biquiver (with relations), reduce this biquiver.