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.
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.
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.
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.
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.
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.
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).
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.
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 ).
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.
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.
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.
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.
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.
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.
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.)
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).
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.
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).
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 .
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!
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.
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 .
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
(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.
(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.
(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.)
(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.
(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.
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.
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.
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.
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.
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.
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.
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!