Oxford-2013-06-24

 Oxford-2013-06-24

Cellularity of generalized q-Schur algebras

Bb90a2011740dd9ed01f00c96ba04259?s=128

Stephen Doty

June 18, 2013
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  1. 2.

    Summary Generalized Schur algebras were introduced by S. Donkin in

    the 1980s. They are determined by a root datum (i.e., a reductive algebraic group or Lie algebra) and a finite saturated set π of dominant weights. They, and their q-analogues, can now be defined in terms of generators and relations.
  2. 3.

    Summary Generalized Schur algebras were introduced by S. Donkin in

    the 1980s. They are determined by a root datum (i.e., a reductive algebraic group or Lie algebra) and a finite saturated set π of dominant weights. They, and their q-analogues, can now be defined in terms of generators and relations. I will discuss a recent preprint [arXiv: 1012.5983] — joint with Giaquinto — in which we derive the cellular structure of these algebras solely from their defining presentation.
  3. 4.

    Summary Generalized Schur algebras were introduced by S. Donkin in

    the 1980s. They are determined by a root datum (i.e., a reductive algebraic group or Lie algebra) and a finite saturated set π of dominant weights. They, and their q-analogues, can now be defined in terms of generators and relations. I will discuss a recent preprint [arXiv: 1012.5983] — joint with Giaquinto — in which we derive the cellular structure of these algebras solely from their defining presentation. This provides a new proof (independent of quantum or algebraic group theory) of the important fact that specializations of generalized q-Schur algebras over a field (of characteristic zero) are quasihereditary.
  4. 5.

    Root datum (of finite type) A root datum is essentially

    just a realization of a Cartan matrix. One might prefer to use a (valued) graph approach instead.
  5. 6.

    Root datum (of finite type) A root datum is essentially

    just a realization of a Cartan matrix. One might prefer to use a (valued) graph approach instead. Recall how root data arise naturally in the theory of algebraic groups.
  6. 7.

    Root datum (of finite type) A root datum is essentially

    just a realization of a Cartan matrix. One might prefer to use a (valued) graph approach instead. Recall how root data arise naturally in the theory of algebraic groups. Suppose we have a (split and connected) reductive algebraic group, G.
  7. 8.

    Root datum (of finite type) A root datum is essentially

    just a realization of a Cartan matrix. One might prefer to use a (valued) graph approach instead. Recall how root data arise naturally in the theory of algebraic groups. Suppose we have a (split and connected) reductive algebraic group, G. Fix a maximal torus T < G. So T ∼ = Gr m , for some r. Have two finitely-generated free abelian groups X ∼ = Zr , X∨ ∼ = Zr defined by: X := X(T) = Hom(T,Gm) = character group of T X∨ := X∨(T) = Hom(Gm,T) = co-character group of T.
  8. 9.

    Root datum (of finite type) A root datum is essentially

    just a realization of a Cartan matrix. One might prefer to use a (valued) graph approach instead. Recall how root data arise naturally in the theory of algebraic groups. Suppose we have a (split and connected) reductive algebraic group, G. Fix a maximal torus T < G. So T ∼ = Gr m , for some r. Have two finitely-generated free abelian groups X ∼ = Zr , X∨ ∼ = Zr defined by: X := X(T) = Hom(T,Gm) = character group of T X∨ := X∨(T) = Hom(Gm,T) = co-character group of T. There is a natural dual pairing , : X ×X∨ → Z which induces an isomorphism X ∼ = Hom Z (X∨,Z).
  9. 10.

    Root datum, continued Since T acts semisimply on any (rational)

    T-module V , we have a weight space decomposition V = λ∈X V λ where V λ = {v ∈ V | t ·v = λ(t)v, all t ∈ T}.
  10. 11.

    Root datum, continued Since T acts semisimply on any (rational)

    T-module V , we have a weight space decomposition V = λ∈X V λ where V λ = {v ∈ V | t ·v = λ(t)v, all t ∈ T}. In particular, applying this fact to the adjoint representation of G in its Lie algebra Lie(G) we have LieG = LieT ⊕ α∈R (LieG)α where R = R(G,T) is the set of non-zero weights for LieG. (So (LieG)0 = LieT.) The set R is called the set of roots or the root system.
  11. 12.

    Root datum, continued Let R∨ = {α∨ | α ∈

    R} < X be the set of coroots. One can define α∨ ∈ X∨(T) by picking an appropriate homomorphism ϕα : SL2 → G with ϕα ( 0 a −a−1 0 ) ∈ NG (T). Then α∨(a) = ϕα (a 0 0 a−1 ).
  12. 13.

    Root datum, continued Let R∨ = {α∨ | α ∈

    R} < X be the set of coroots. One can define α∨ ∈ X∨(T) by picking an appropriate homomorphism ϕα : SL2 → G with ϕα ( 0 a −a−1 0 ) ∈ NG (T). Then α∨(a) = ϕα (a 0 0 a−1 ). The Weyl group W is the subgroup of GL(X) generated by all sα , where sα (λ) = λ − λ,α∨ α for any λ ∈ X. Regard W as a Coxeter group by extending each sα to a reflection on the Euclidean space R⊗ Z X.
  13. 14.

    Root datum, continued Let R∨ = {α∨ | α ∈

    R} < X be the set of coroots. One can define α∨ ∈ X∨(T) by picking an appropriate homomorphism ϕα : SL2 → G with ϕα ( 0 a −a−1 0 ) ∈ NG (T). Then α∨(a) = ϕα (a 0 0 a−1 ). The Weyl group W is the subgroup of GL(X) generated by all sα , where sα (λ) = λ − λ,α∨ α for any λ ∈ X. Regard W as a Coxeter group by extending each sα to a reflection on the Euclidean space R⊗ Z X. Pick a set Π of simple roots in R. This amounts to choosing a positive Borel subgroup B+ with T < B+ < G. Let Π∨ = {α∨ | α ∈ Π} be the corresponding set of simple coroots.
  14. 15.

    Root datum, continued Let E(R) = the subspace of R⊗

    Z X generated by R. If R is irreducible then there is a unique W -invariant inner product ( , ) on E(R) such that (α,α) = 2 for all short roots α ∈ R.
  15. 16.

    Root datum, continued Let E(R) = the subspace of R⊗

    Z X generated by R. If R is irreducible then there is a unique W -invariant inner product ( , ) on E(R) such that (α,α) = 2 for all short roots α ∈ R. In general, R is a disjoint union of finitely many irreducible components, each of which has a unique W -invariant inner product as above. Taking the orthogonal sum of these inner products, we obtain a W -invariant inner product ( , ) on E(R) such that the short roots α in any irreducible component satisfy (α,α) = 2. Then a α,β := β,α∨ = 2(β,α) (α,α) for all α,β ∈ R. The integral matrix (a α,β ) α,β∈Π is the symmetrizable Cartan matrix associated to the root datum.
  16. 17.

    Root datum, continued A root datum is any quadruple (X,Π,X∨,Π∨)

    arising as above (one can give axioms). Actually, this is a root datum of finite type; more general root data arise in the theory of Kac–Moody groups and Lie algebras, but for this talk we consider only those of finite type.
  17. 18.

    Root datum, continued A root datum is any quadruple (X,Π,X∨,Π∨)

    arising as above (one can give axioms). Actually, this is a root datum of finite type; more general root data arise in the theory of Kac–Moody groups and Lie algebras, but for this talk we consider only those of finite type. Theorem (classification) Any reductive algebraic group G has a root datum. Conversely, any root datum determines a unique reductive algebraic group G, up to isomorphism.
  18. 19.

    Root datum, continued A root datum is any quadruple (X,Π,X∨,Π∨)

    arising as above (one can give axioms). Actually, this is a root datum of finite type; more general root data arise in the theory of Kac–Moody groups and Lie algebras, but for this talk we consider only those of finite type. Theorem (classification) Any reductive algebraic group G has a root datum. Conversely, any root datum determines a unique reductive algebraic group G, up to isomorphism. To every root datum there is an associated reductive Lie algebra g and an associated quantized enveloping algebra U = U(g); see e.g., Lusztig’s book.
  19. 20.

    Saturated sets of weights Fix a root datum (X,Π,X∨,Π∨) as

    above. Recall the usual dominance order ≥ on X defined by λ ≥ µ ⇐⇒ λ − µ ∈ ∑α∈Π Z≥0α. Let X+ = {λ ∈ X | λ,α∨ ≥ 0, all α ∈ Π} be the usual set of dominant weights. (These label the simple G-modules.)
  20. 21.

    Saturated sets of weights Fix a root datum (X,Π,X∨,Π∨) as

    above. Recall the usual dominance order ≥ on X defined by λ ≥ µ ⇐⇒ λ − µ ∈ ∑α∈Π Z≥0α. Let X+ = {λ ∈ X | λ,α∨ ≥ 0, all α ∈ Π} be the usual set of dominant weights. (These label the simple G-modules.) Definition A set π ⊂ X+ is saturated if π is predecessor-closed (i.e., ∀λ ∈ X+, ∀µ ∈ π, λ ≤ µ =⇒ λ ∈ π).
  21. 22.

    Saturated sets of weights Fix a root datum (X,Π,X∨,Π∨) as

    above. Recall the usual dominance order ≥ on X defined by λ ≥ µ ⇐⇒ λ − µ ∈ ∑α∈Π Z≥0α. Let X+ = {λ ∈ X | λ,α∨ ≥ 0, all α ∈ Π} be the usual set of dominant weights. (These label the simple G-modules.) Definition A set π ⊂ X+ is saturated if π is predecessor-closed (i.e., ∀λ ∈ X+, ∀µ ∈ π, λ ≤ µ =⇒ λ ∈ π). Saturated sets exist in abundance: X+[≤ µ] = {λ ∈ X+ | λ ≤ µ} is saturated, for any given µ ∈ X+. Any union of such sets is again saturated.
  22. 23.

    q-notation Let v be an indeterminate, and Q(v) the field

    of rational functions over v. For any k ∈ Z we have the quantum integer [k] defined by [k] = vk −v−k v −v−1 = vk−1 +vk−3 +···+v−k+3 +v−k+1. For any n,k ∈ Z such that k > 0 we define quantum factorials [k]! and quantum binomial coefficients n k by [k]! = [k][k −1]···[2][1]; [0]! = 1 n k = [n][n −1]···[n −k +1] [k]! ; n 0 = 1. All of these elements belong to the subring Z[v,v−1] ⊂ Q(v).
  23. 25.

    Modification for handling non simply-laced cases Write dα = (α,α)

    2 ∈ {1,2,3} for any α ∈ Π. Put vα := vdα .
  24. 26.

    Modification for handling non simply-laced cases Write dα = (α,α)

    2 ∈ {1,2,3} for any α ∈ Π. Put vα := vdα . More generally, given any rational function F = F(v) ∈ Q(v), put Fα := F(v)|v=vα . In particular, we have the notations [n]α , [k]! α , and n k α obtained from the corresponding elements [n], [k]!, and n k (for k ≥ 0) by replacing all occurrences of the variable v by the power vα = vdα .
  25. 27.

    Definition (Rational form of generalized q-Schur algebra) Let π be

    a finite saturated subset of X+. Let S(π) be the associative algebra with 1 over Q(v) defined by generators Eα , Fα , 1 λ (α ∈ Π, λ ∈ W π) subject to the relations (R1) 1 λ 1µ = δλµ 1 λ ; ∑λ∈W π 1 λ = 1 (R2) Eα F β −F β Eα = δαβ ∑λ∈W π [ λ,α∨ ]α 1 λ (R3) Eα 1 λ = 1 λ+α Eα , 1 λ Eα = Eα 1 λ−α , Fα 1 λ = 1 λ−α Fα , 1 λ Fα = Fα 1 λ+α with the stipulation that 1 λ±α = 0 whenever λ ±α / ∈ W π.
  26. 28.

    Definition (Rational form of generalized q-Schur algebra) Let π be

    a finite saturated subset of X+. Let S(π) be the associative algebra with 1 over Q(v) defined by generators Eα , Fα , 1 λ (α ∈ Π, λ ∈ W π) subject to the relations (R1) 1 λ 1µ = δλµ 1 λ ; ∑λ∈W π 1 λ = 1 (R2) Eα F β −F β Eα = δαβ ∑λ∈W π [ λ,α∨ ]α 1 λ (R3) Eα 1 λ = 1 λ+α Eα , 1 λ Eα = Eα 1 λ−α , Fα 1 λ = 1 λ−α Fα , 1 λ Fα = Fα 1 λ+α with the stipulation that 1 λ±α = 0 whenever λ ±α / ∈ W π. This comes from [D, Rep. Theory 2003]. Discovered earlier in type A in [D–Giaquito, IMRN, 2002].
  27. 29.

    Definition (Rational form of generalized q-Schur algebra) Let π be

    a finite saturated subset of X+. Let S(π) be the associative algebra with 1 over Q(v) defined by generators Eα , Fα , 1 λ (α ∈ Π, λ ∈ W π) subject to the relations (R1) 1 λ 1µ = δλµ 1 λ ; ∑λ∈W π 1 λ = 1 (R2) Eα F β −F β Eα = δαβ ∑λ∈W π [ λ,α∨ ]α 1 λ (R3) Eα 1 λ = 1 λ+α Eα , 1 λ Eα = Eα 1 λ−α , Fα 1 λ = 1 λ−α Fα , 1 λ Fα = Fα 1 λ+α with the stipulation that 1 λ±α = 0 whenever λ ±α / ∈ W π. This comes from [D, Rep. Theory 2003]. Discovered earlier in type A in [D–Giaquito, IMRN, 2002]. In the above papers, the q-Serre relations were included in the presentation, but (as pointed out by R. Rouquier) the q-Serre relations are superfluous!
  28. 30.

    A-form and k-form Let A = Q[v,v−1]. Let S(π) A

    be the A-subalgebra of S(π) generated by all q-divided powers Eα [k]! α , Fα [k]! α (α ∈ Π) along with the idempotents 1 λ (λ ∈ W π).
  29. 31.

    A-form and k-form Let A = Q[v,v−1]. Let S(π) A

    be the A-subalgebra of S(π) generated by all q-divided powers Eα [k]! α , Fα [k]! α (α ∈ Π) along with the idempotents 1 λ (λ ∈ W π). Since A is a PID, and S(π) A is torsion-free over A, it follows that S(π) A is free as an A-module.
  30. 32.

    A-form and k-form Let A = Q[v,v−1]. Let S(π) A

    be the A-subalgebra of S(π) generated by all q-divided powers Eα [k]! α , Fα [k]! α (α ∈ Π) along with the idempotents 1 λ (λ ∈ W π). Since A is a PID, and S(π) A is torsion-free over A, it follows that S(π) A is free as an A-module. Definition (k-form S(π) k of S(π)) Let k be a field of characteristic zero, and 0 = v ∈ k. Regard k as an A-algebra via the morphism A → k such that v → v. Put S(π) k = k⊗ A S(π) A .
  31. 33.

    Example (The Dipper–James q-Schur algebra Sq(n,r)) Let the underlying root

    datum be that of G = GLn. Let π be the set of partitions of r into no more than n parts. Choose 0 = v ∈ k and put q = v2. Then S(π) k ∼ = Sq(n,r), the original q-Schur algebra in type A defined by Dipper and James.
  32. 34.

    Example (The Dipper–James q-Schur algebra Sq(n,r)) Let the underlying root

    datum be that of G = GLn. Let π be the set of partitions of r into no more than n parts. Choose 0 = v ∈ k and put q = v2. Then S(π) k ∼ = Sq(n,r), the original q-Schur algebra in type A defined by Dipper and James. The definition given by Dipper and James used the Iwahori–Hecke algebra Hq(Sr ) given by generators Ti (i = 1,...,r −1) and relations (a) (Ti −q)(Ti +1) = 0 (b) Ti Ti+1Ti = Ti+1Ti Ti+1 (c) Ti Tj = Tj Ti if |i −j| > 1. When one specializes q → 1, Hq(Sr ) ∼ = kSr (group algebra of the symmetric group). Hence when q → 1, we have Sq(n,r) ∼ = S(n,r), the classical Schur algebra in type A.
  33. 35.

    Theorem (Main Result [arXiv: 1012.5983]) Let k be a field

    of characteristic 0, regarded as A-algebra via the specialization v → v, for 0 = v ∈ k. Then:
  34. 36.

    Theorem (Main Result [arXiv: 1012.5983]) Let k be a field

    of characteristic 0, regarded as A-algebra via the specialization v → v, for 0 = v ∈ k. Then: (a) S(π) A and S(π) k are cellular algebras.
  35. 37.

    Theorem (Main Result [arXiv: 1012.5983]) Let k be a field

    of characteristic 0, regarded as A-algebra via the specialization v → v, for 0 = v ∈ k. Then: (a) S(π) A and S(π) k are cellular algebras. (b) For each λ ∈ π, there is a simple S(π) k -module of highest weight λ.
  36. 38.

    Theorem (Main Result [arXiv: 1012.5983]) Let k be a field

    of characteristic 0, regarded as A-algebra via the specialization v → v, for 0 = v ∈ k. Then: (a) S(π) A and S(π) k are cellular algebras. (b) For each λ ∈ π, there is a simple S(π) k -module of highest weight λ. (c) S(π) k is a quasihereditary algebra.
  37. 39.

    Theorem (Main Result [arXiv: 1012.5983]) Let k be a field

    of characteristic 0, regarded as A-algebra via the specialization v → v, for 0 = v ∈ k. Then: (a) S(π) A and S(π) k are cellular algebras. (b) For each λ ∈ π, there is a simple S(π) k -module of highest weight λ. (c) S(π) k is a quasihereditary algebra. Part (c) is an immediate consequence of parts (a), (b).
  38. 40.

    Theorem (Main Result [arXiv: 1012.5983]) Let k be a field

    of characteristic 0, regarded as A-algebra via the specialization v → v, for 0 = v ∈ k. Then: (a) S(π) A and S(π) k are cellular algebras. (b) For each λ ∈ π, there is a simple S(π) k -module of highest weight λ. (c) S(π) k is a quasihereditary algebra. Part (c) is an immediate consequence of parts (a), (b). Proof uses only the defining presentation for S(π), along with a few basic facts (e.g., Weyl’s theorem) about semisimple Lie algebras. In particular, no facts from the theory of quantum groups are used in the proof.
  39. 41.

    Theorem (Main Result [arXiv: 1012.5983]) Let k be a field

    of characteristic 0, regarded as A-algebra via the specialization v → v, for 0 = v ∈ k. Then: (a) S(π) A and S(π) k are cellular algebras. (b) For each λ ∈ π, there is a simple S(π) k -module of highest weight λ. (c) S(π) k is a quasihereditary algebra. Part (c) is an immediate consequence of parts (a), (b). Proof uses only the defining presentation for S(π), along with a few basic facts (e.g., Weyl’s theorem) about semisimple Lie algebras. In particular, no facts from the theory of quantum groups are used in the proof. Proved originally in [D, Rep. Theory 2003], but that proof used properties of the Lusztig–Kashiwara canonical basis.
  40. 42.

    Theorem (Main Result [arXiv: 1012.5983]) Let k be a field

    of characteristic 0, regarded as A-algebra via the specialization v → v, for 0 = v ∈ k. Then: (a) S(π) A and S(π) k are cellular algebras. (b) For each λ ∈ π, there is a simple S(π) k -module of highest weight λ. (c) S(π) k is a quasihereditary algebra. Part (c) is an immediate consequence of parts (a), (b). Proof uses only the defining presentation for S(π), along with a few basic facts (e.g., Weyl’s theorem) about semisimple Lie algebras. In particular, no facts from the theory of quantum groups are used in the proof. Proved originally in [D, Rep. Theory 2003], but that proof used properties of the Lusztig–Kashiwara canonical basis. We would like to replace A = Q[v,v−1] by the ring Z[v,v−1]. Then k could be any field.
  41. 43.

    Outline of proof 1. If π is a saturated subset

    of π then there is a unique surjective algebra homomorphism pπ,π : S(π) → S(π ) such that Eα → Eα , Fα → Fα and 1 λ → 1 λ or zero (if λ / ∈ W π ). This makes it possible to proceed by induction.
  42. 44.

    Outline of proof 1. If π is a saturated subset

    of π then there is a unique surjective algebra homomorphism pπ,π : S(π) → S(π ) such that Eα → Eα , Fα → Fα and 1 λ → 1 λ or zero (if λ / ∈ W π ). This makes it possible to proceed by induction. 2. There is a unique Q(v)-linear algebra anti-involution ∗ on S(π) such that E∗ α = Fα , F∗ α = Eα , and 1∗ λ = 1 λ . This is easy to verify from the defining relations.
  43. 45.

    Outline of proof 1. If π is a saturated subset

    of π then there is a unique surjective algebra homomorphism pπ,π : S(π) → S(π ) such that Eα → Eα , Fα → Fα and 1 λ → 1 λ or zero (if λ / ∈ W π ). This makes it possible to proceed by induction. 2. There is a unique Q(v)-linear algebra anti-involution ∗ on S(π) such that E∗ α = Fα , F∗ α = Eα , and 1∗ λ = 1 λ . This is easy to verify from the defining relations. 3. Put E(a) α := Eα a [a]! α , F(a) α := Fα a [a]! α . Then from (R3) it follows that E(a) α 1 λ = 1 λ+aα E(a) α , F(a) α 1 λ = 1 λ−aα F(a) α , 1 λ E(a) α = E(a) α 1 λ−aα , 1 λ F(a) α = F(a) α 1 λ+aα where we adopt the convention that the symbol 1µ = 0 whenever µ / ∈ W π.
  44. 46.

    Outline of proof 4. By a double induction on a,

    b one proves using the defining relations (R1)–(R3) that E(a) α F(b) α 1 λ = ∑ t≥0 a−b + λ,α∨ t α F(b−t) α E(a−t) α 1 λ F(b) α E(a) α 1 λ = ∑ t≥0 b −a− λ,α∨ t α E(a−t) α F(b−t) α 1 λ for λ ∈ W π, α a simple root, a,b ≥ 0.
  45. 47.

    Outline of proof 4. By a double induction on a,

    b one proves using the defining relations (R1)–(R3) that E(a) α F(b) α 1 λ = ∑ t≥0 a−b + λ,α∨ t α F(b−t) α E(a−t) α 1 λ F(b) α E(a) α 1 λ = ∑ t≥0 b −a− λ,α∨ t α E(a−t) α F(b−t) α 1 λ for λ ∈ W π, α a simple root, a,b ≥ 0. 5. Fix π. S = S(π) has a “triangular” decomposition S = S−S0S+, where S− = Fα : α ∈ Π , S+ = Eα : α ∈ Π , S0 = 1 λ : λ ∈ W π . The same decomposition holds if the factors on the RHS are permuted.
  46. 48.

    Outline of proof 4. By a double induction on a,

    b one proves using the defining relations (R1)–(R3) that E(a) α F(b) α 1 λ = ∑ t≥0 a−b + λ,α∨ t α F(b−t) α E(a−t) α 1 λ F(b) α E(a) α 1 λ = ∑ t≥0 b −a− λ,α∨ t α E(a−t) α F(b−t) α 1 λ for λ ∈ W π, α a simple root, a,b ≥ 0. 5. Fix π. S = S(π) has a “triangular” decomposition S = S−S0S+, where S− = Fα : α ∈ Π , S+ = Eα : α ∈ Π , S0 = 1 λ : λ ∈ W π . The same decomposition holds if the factors on the RHS are permuted. 6. S is finite dimensional over Q(v), and each Eα , Fα is nilpotent.
  47. 49.

    Outline of proof 7. S = ∑λ∈π S−1 λ S+.

    (Need that 1w(λ) ∈ S1 λ S (mod lower terms) for any w ∈ W .)
  48. 50.

    Outline of proof 7. S = ∑λ∈π S−1 λ S+.

    (Need that 1w(λ) ∈ S1 λ S (mod lower terms) for any w ∈ W .) 8. For a subset Φ ⊆ π, define S[Φ] = ∑λ∈Φ S−1 λ S+. If Φ is cosaturated (successor-closed) in π then S[Φ] is a two-sided ideal in S.
  49. 51.

    Outline of proof 7. S = ∑λ∈π S−1 λ S+.

    (Need that 1w(λ) ∈ S1 λ S (mod lower terms) for any w ∈ W .) 8. For a subset Φ ⊆ π, define S[Φ] = ∑λ∈Φ S−1 λ S+. If Φ is cosaturated (successor-closed) in π then S[Φ] is a two-sided ideal in S. 9. In particular, (i) S[≥ λ] and S[> λ] are two-sided ideals, for any λ ∈ π.
  50. 52.

    Outline of proof 7. S = ∑λ∈π S−1 λ S+.

    (Need that 1w(λ) ∈ S1 λ S (mod lower terms) for any w ∈ W .) 8. For a subset Φ ⊆ π, define S[Φ] = ∑λ∈Φ S−1 λ S+. If Φ is cosaturated (successor-closed) in π then S[Φ] is a two-sided ideal in S. 9. In particular, (i) S[≥ λ] and S[> λ] are two-sided ideals, for any λ ∈ π. (ii) If λ ∈ π is maximal with respect to ≥ then {λ} is cosaturated in π, and S[{λ}] = S−1 λ S+ = S1 λ S is a two-sided ideal. (Specialize v → 1 and apply Weyl’s theorem for ss Lie algebras.)
  51. 53.

    Outline of proof 7. S = ∑λ∈π S−1 λ S+.

    (Need that 1w(λ) ∈ S1 λ S (mod lower terms) for any w ∈ W .) 8. For a subset Φ ⊆ π, define S[Φ] = ∑λ∈Φ S−1 λ S+. If Φ is cosaturated (successor-closed) in π then S[Φ] is a two-sided ideal in S. 9. In particular, (i) S[≥ λ] and S[> λ] are two-sided ideals, for any λ ∈ π. (ii) If λ ∈ π is maximal with respect to ≥ then {λ} is cosaturated in π, and S[{λ}] = S−1 λ S+ = S1 λ S is a two-sided ideal. (Specialize v → 1 and apply Weyl’s theorem for ss Lie algebras.) 10. If λ ∈ π is maximal then ∆(λ) = S−1 λ = S1 λ is a cell module for S. Moreover, S1 λ S ∼ = ∆(λ)⊗ Q(v) ∆(λ)∗ as bimodules.
  52. 54.

    Outline of proof 7. S = ∑λ∈π S−1 λ S+.

    (Need that 1w(λ) ∈ S1 λ S (mod lower terms) for any w ∈ W .) 8. For a subset Φ ⊆ π, define S[Φ] = ∑λ∈Φ S−1 λ S+. If Φ is cosaturated (successor-closed) in π then S[Φ] is a two-sided ideal in S. 9. In particular, (i) S[≥ λ] and S[> λ] are two-sided ideals, for any λ ∈ π. (ii) If λ ∈ π is maximal with respect to ≥ then {λ} is cosaturated in π, and S[{λ}] = S−1 λ S+ = S1 λ S is a two-sided ideal. (Specialize v → 1 and apply Weyl’s theorem for ss Lie algebras.) 10. If λ ∈ π is maximal then ∆(λ) = S−1 λ = S1 λ is a cell module for S. Moreover, S1 λ S ∼ = ∆(λ)⊗ Q(v) ∆(λ)∗ as bimodules. 11. Apply induction on π (noting that π = π −{λ} is saturated and so defines a smaller algebra S(π )), and use Koenig and Xi’s basis-free reformulation of Graham and Lehrer’s original definition of cellular algebra.
  53. 55.

    Outline of proof 12. Check that the cell structure is

    compatible with restriction to the A-form generated by the E(a) α , F(b) α , and 1 λ .
  54. 56.

    Outline of proof 12. Check that the cell structure is

    compatible with restriction to the A-form generated by the E(a) α , F(b) α , and 1 λ . Remarks (The rank 1 case) (a) It is necessary to work out the representations of S explicitly in the rank 1 case. This is easy.
  55. 57.

    Outline of proof 12. Check that the cell structure is

    compatible with restriction to the A-form generated by the E(a) α , F(b) α , and 1 λ . Remarks (The rank 1 case) (a) It is necessary to work out the representations of S explicitly in the rank 1 case. This is easy. (b) Assume the number of simple roots is ≥ 2. Fix a simple root α. We define a rank 1 subalgebra Sα of S = S(π) and consider the “adjoint” action of Sα on S. The rep theory in rank 1 then implies the q-Serre relations hold in S.
  56. 58.

    Outline of proof 12. Check that the cell structure is

    compatible with restriction to the A-form generated by the E(a) α , F(b) α , and 1 λ . Remarks (The rank 1 case) (a) It is necessary to work out the representations of S explicitly in the rank 1 case. This is easy. (b) Assume the number of simple roots is ≥ 2. Fix a simple root α. We define a rank 1 subalgebra Sα of S = S(π) and consider the “adjoint” action of Sα on S. The rep theory in rank 1 then implies the q-Serre relations hold in S. (c) The explicit form of the rep theory in the rank one case leads to an explicit cellular basis of the algebra. A slight variant of it coincides with the canonical basis.
  57. 59.

    Let A = Z[v,v−1]. Let S(π)A = the A -subalgebra

    of S(π) generated by all q-divided powers E(a) α , F(b) α along with the idempotents 1 λ .
  58. 60.

    Let A = Z[v,v−1]. Let S(π)A = the A -subalgebra

    of S(π) generated by all q-divided powers E(a) α , F(b) α along with the idempotents 1 λ . QUESTION: Is S(π)A free as an A -module?
  59. 61.

    Let A = Z[v,v−1]. Let S(π)A = the A -subalgebra

    of S(π) generated by all q-divided powers E(a) α , F(b) α along with the idempotents 1 λ . QUESTION: Is S(π)A free as an A -module? Enough to show that each ∆(λ)A = S(π)A 1 λ is free as an A -module, where λ is a maximal element of π.
  60. 62.

    Let A = Z[v,v−1]. Let S(π)A = the A -subalgebra

    of S(π) generated by all q-divided powers E(a) α , F(b) α along with the idempotents 1 λ . QUESTION: Is S(π)A free as an A -module? Enough to show that each ∆(λ)A = S(π)A 1 λ is free as an A -module, where λ is a maximal element of π. This would enable us to extend our main result to any ground field k.
  61. 63.

    Let A = Z[v,v−1]. Let S(π)A = the A -subalgebra

    of S(π) generated by all q-divided powers E(a) α , F(b) α along with the idempotents 1 λ . QUESTION: Is S(π)A free as an A -module? Enough to show that each ∆(λ)A = S(π)A 1 λ is free as an A -module, where λ is a maximal element of π. This would enable us to extend our main result to any ground field k. Such an extension is known [D, Rep. Theory 2003] but we are looking for an elementary proof which does not use the theory of quantum groups, in keeping with our elementary approach to generalized q-Schur algebras via the defining presentation given in this talk.
  62. 64.

    Let A = Z[v,v−1]. Let S(π)A = the A -subalgebra

    of S(π) generated by all q-divided powers E(a) α , F(b) α along with the idempotents 1 λ . QUESTION: Is S(π)A free as an A -module? Enough to show that each ∆(λ)A = S(π)A 1 λ is free as an A -module, where λ is a maximal element of π. This would enable us to extend our main result to any ground field k. Such an extension is known [D, Rep. Theory 2003] but we are looking for an elementary proof which does not use the theory of quantum groups, in keeping with our elementary approach to generalized q-Schur algebras via the defining presentation given in this talk. Thanks for listening!