Cellular structures using U q -tilting modules

Transcription

1 Cellular structures using U q -tilting modules Or: centralizer algebras are fun! Daniel Tubbenhauer q(λ) g λ i M f λ j ι λ T q(λ) N g λ i f λ j π λ q(λ) Joint work with Henning Haahr Andersen and Catharina Stroppel April 2015 Daniel Tubbenhauer April / 26

2 The main theorem Theorem Let T be a U q = U q (g)-tilting module. Then End Uq (T ) is a cellular algebra. I have to explain the words in red. But let us start with an example. Example(Schur 1901) Let K[S d ] be the symmetric group in d letters and let 1 (ω 1 ) be the vector representation of U 1 = U 1 (gl n ). Take T = 1 (ω 1 ) d, then Φ SW : K[S d ] End U1 (T ) and Φ SW : K[S d ] = End U1 (T ), if n d. Since T is a U 1 -tilting module, K[S d ] is cellular. Daniel Tubbenhauer April / 26

3 1 U q -tilting modules U q and its representation theory The category of U q -tilting modules 2 Cellularity of End Uq (T ) Cellular algebras Cellularity and U q -tilting modules 3 The representation theory of End Uq (T ) Consequences of cellularity - U q -tilting view Examples that fit into the picture Daniel Tubbenhauer April / 26

4 Quantum groups at roots of unity Fix an arbitrary element q K {0}. Define U q = U q (g) = U A A K. Here U A = U A (g) is Lusztig s A-form: the A-subalgebra of U v = U v (g) generated by K ±1 i, E (j) i and F (j) i for i = 1,..., n 1 and j N. Example In the sl 2 case, the Q(v)-algebra U v (sl 2 ) is generated by K, K 1 and E, F subject to some relations. Let q be a complex, primitive third root of unity. U q (sl 2 ) is generated by K, K 1, E, F, E (3) and F (3) subject to some relations. Here E (3), F (3) are extra generators, since E 3 = [3]!E (3) = 0 because of [3] = 0. Daniel Tubbenhauer Uq and its representation theory April / 26

5 Weyl modules as building blocks For each dominant U v -weight λ X + there is a simple U v -module v (λ) called Weyl module. Fact: the set { v (λ) λ X + } is a complete set of pairwise non-isomorphic, simple U v -modules (of type 1). Example For sl 2 we have X + = Z 0. The Weyl module v (3) is v 3 [1] m 3 [3] v 1 m 2 [2] [2] v +1 [3] m 1 [1] v +3 m 0, where E acts to the right, F acts to the left and K acts as a loop. The category of finite dimensional U v -modules is semi-simple. Daniel Tubbenhauer Uq and its representation theory April / 26

6 Weyl modules as building blocks? Fact: the q (λ) s are no longer (semi-)simple in general. But they have unique simple heads L q (λ). Fact: the set {L q (λ) λ X + } is a complete set of pairwise non-isomorphic, simple U q -modules (of type 1). Example Let g = sl 2 and q be a complex, primitive third root of unity. q (3) is q 3 m q 1 m q +1 0 m 1 m q +3 The C-span of {m 1, m 2 } is now stable under the action of U q (sl 2 ): this is L q (1). The simple head is L q (3) = q (3)/L q (1) and is spanned by {m 0, m 3 }. The category of finite dimensional U q -modules is not semi-simple in general. Daniel Tubbenhauer Uq and its representation theory April / 26

7 U q -tilting modules as building blocks? Let q (λ) be a Weyl module and q (λ) its dual. A U q -tilting module T is a U q -module with a q -filtration and a q -filtration: T = M 0 M 1 M k M k 1 M k = 0, 0 = N 0 N 1 N k N k 1 N k = T, such that M k /M k +1 is some q (λ) and N k +1/N k is some q (λ). Example All U v -modules are U v -tilting modules. For our favorite example q 3 = 1 C and g = sl 2 : q (i) is a U q -tilting module iff i = 0, 1 or i 1 mod 3. Daniel Tubbenhauer The category of Uq -tilting modules April / 26

8 U q -tilting modules as building blocks. The category of U q -tilting modules T has some nice properties: T is closed under finite tensor products. The indecomposables T q (λ) of T are parametrized by λ X +. They have λ as their maximal weight and contain q (λ) with multiplicity 1. We have q (λ) ι λ T q (λ) π λ q (λ). Example The vector representation q (1) is a U q (sl 2 )-tilting module. Thus, T = q (1) d is. Then T q (d) is the indecomposable summand of T containing q (d). Example q (λ) is a U q -tilting module for minuscule λ. Thus, tensor products of these are. Daniel Tubbenhauer The category of Uq -tilting modules April / 26

9 The Ext-vanishing We have for all λ, µ X + that Ext i U q ( q (λ), q (µ)) = { Kc λ, if i = 0 and λ = µ, 0, else, where c λ : q (λ) q (λ) is the U q -homomorphisms that sends head to socle. Assume that M has a q -filtration and N has a q -filtration. We have dim(hom Uq (M, q (λ))) = (M : q (λ)). We have dim(hom Uq ( q (λ), N)) = (N : q (λ)). Daniel Tubbenhauer The category of Uq -tilting modules April / 26

10 U q -tilting modules as building blocks! T T iff Ext 1 U q (T, q (λ)) = 0 = Ext 1 U q ( q (λ), T ) for all λ X +. In particular, if M has a q - and N has a q -filtration: q (λ) ι λ M f λ j T q (λ) g λ i g λ i N f λ j π λ q (λ) In words: any U q -homomorphism g : q (λ) N extends to an U q -homomorphism g : T q (λ) N whereas any U q -homomorphism f : M q (λ) factors through T q (λ) via f : M T q (λ). Daniel Tubbenhauer The category of Uq -tilting modules April / 26

11 Exempli gratia Consequence of the discussion before: dim(end Uq (T )) = λ X + (T : q (λ)) 2 = λ X + (T : q (λ)) 2. Take T = q (λ) d. If λ X + is minuscule as a U q -weight, then q (λ) is always U q -tilting and dim(end Uq (T )) is independent of K and q, since q (λ) has a character independent of K and of q. Example By quantum Schur-Weyl, we see that Φ qsw : H d (q) End Uq (T ) and Φ qsw : H d (q) = End Uq (T ), if n d. Thus, dim(h d (q)) independent of K and q. Daniel Tubbenhauer The category of Uq -tilting modules April / 26

12 Exempli gratia (Temperley-Lieb without diagrams) Let us consider our favorite case again. From the construction of T q (3): q (3) T q (3) q (1). We compute: T v = v (1) v (1) v (1) = v (3) v (1) v (1), whereas T q = q (1) q (1) q (1) = T q (3) T q (1). In particular, dim(end Uv (sl 2)(T v )) = dim(end Uq(sl 2)(T q )) = = 5. Note that End Uq(sl 2)( q (1) d ) is the Temperley-Lieb algebra T L d (δ). Daniel Tubbenhauer The category of Uq -tilting modules April / 26

13 Cellular algebras Definition(Graham-Lehrer 1996) A K-algebra A is cellular if it has a basis {c λ ij λ P, i, j I}, where (P, ) is a finite poset and I λ is a finite set, such that 1 The map i: A A, c λ ij c λ ji is an anti-isomorphism. 2 We have (for friend of higher order) ac λ ij = k I λ r ik (a)c λ kj + friends. Note that the scalars r ik (a) do not depend on j. Thus, we think of the basis elements as having independent bottom and top parts. Daniel Tubbenhauer Cellular algebras April / 26

14 Prototype of a cellular basis Example(Specht 1935, Murphy 1995) P = Young diagrams λ, I λ = standard tableaux i, j. c λ ij = P (i) e(λ) P(j) permutation idempotent permutation Form S λ = {c λ j } with formal c λ j and action given by the r ik (a). The set {D λ = S λ /Rad(S λ ) λ P 0 } forms a complete set of pairwise non-isomorphic, simple K[S d ]-modules. Theorem(Graham-Lehrer 1996) This works in general for cellular algebras. Daniel Tubbenhauer Cellular algebras April / 26

15 And for End Uq (T )? Let M have a q - and N have q -filtration. Consider I λ = {1,..., (N : q (λ))} and J λ = {1,..., (M : q (λ))}. By Ext-vanishing, we have diagrams M f λ j T q (λ) f λ j π λ q (λ) and q (λ) ι λ T q (λ) g λ i g λ i N Take any bases F λ = {fj λ : M q (λ) j J λ } of Hom Uq (M, q (λ)) and G λ = {gi λ : q (λ) N i I λ } of Hom Uq ( q (λ), N). Set for each λ X + and all i I λ, j J λ. c λ ij = g λ i f λ j Hom Uq (M, N) Daniel Tubbenhauer Cellularity and Uq -tilting modules April / 26

16 End Uq (T ) is prototypical cellular Cell datum: (P, ) = ({λ X + (T : q (λ)) = (T : q (λ)) 0}, X ). I λ = {1,..., (T : q (λ))} = {1,..., (T : q (λ))} = J λ for each λ P. K-linear anti-involution i: End Uq (T ) End Uq (T ), φ D(φ). Note that D( q (λ)) = q (λ) and D( q (λ)) = q (λ). Cellular basis {cij λ λ P, i, j I λ }. Theorem This gives a cellular datum on End Uq (T ) for any U q -tilting module T. c λ ij = q (λ) Tq (λ) q (λ) filtration part indecomposable filtration part Daniel Tubbenhauer Cellularity and Uq -tilting modules April / 26

17 Exempli gratia (generic Temperley-Lieb) Take K = C and T = v (1) 3 = v (3) l v (1) r v (1). Then P = {1, 3}. We have I 1 = {1, 2} and I 3 = {1}. Thus, we have a basis l v (1) l v (1) r v (1) c 1 11 = v (1) Tv (1) c 1 12 = v (1) Tv (1) c 1 21 = v (1) Tv (1) v (1) v (1) v (1) l v (1) r v (1) l v (1) r v (1) v (3) v (1) v (3) c 1 22 = Tv (1) c 3 11 = Tv (3) v (1) v (3) r v (1) v (3) Daniel Tubbenhauer Cellularity and Uq -tilting modules April / 26

18 Exempli gratia (roots of unity Temperley-Lieb) Take T = q (1) 3 = T q (3) T q (1). Then P = {1, 3}. We have I 1 = {1, 2} and I 3 = {1}. Consider 1 I 1 as indexing the factor q (1) of T q (1) and 2 I 1 the factor q (1) of T q (3). Thus, we have a basis Tq (1) Tq (1) Tq (3) q (1) q (1) q (1) c 1 11 = Tq (1) c 1 12 = Tq (1) c 1 21 = Tq (1) q (1) q (1) q (1) Tq (1) Tq (3) Tq (1) Tq (3) Tq (3) q (1) q (3) c 1 22 = Tq (1) c 3 11 = Tq (3) q (1) q (3) Tq (3) Tq (3) Daniel Tubbenhauer Cellularity and Uq -tilting modules April / 26

19 Cellular pairing and simple End Uq (T )-modules Let T be a U q -tilting module. For λ P define ϑ λ via i(h) g = ϑ λ (g, h)c λ, g, h C(λ) = Hom Uq ( q (λ), T ). Define P 0 = {λ P ϑ λ 0} and Rad(λ) = {g C(λ) ϑ λ (g, C(λ)) = 0}. Theorem(Graham-Lehrer reinterpreted) The set {L(λ) = C(λ)/Rad(λ) λ P 0 } is a complete set of pairwise non-isomorphic, simple End Uq (T )-modules. λ P 0 iff T q (λ) is a summand of T. Moreover, dim(l(λ)) = m λ, T = T q (λ) mλ. λ X + Daniel Tubbenhauer Consequences of cellularity - Uq -tilting view April / 26

20 Exempli gratia (Temperley-Lieb again) Because T v = v (3) v (1) v (1) and T q = Tq (3) T q (1) we see that P 0 = {1, 3} in both cases. In the generic case: C(3) = L(3) = {g 3 1 : v (3) T v }, C(1) = L(1) = {g 1 j : v (1) T v j = 1, 2}, dim(l(3)) = 1 and dim(l(1)) = 2. In the non-semisimple case: C(3) = L(3) = {g 3 1 : q (3) T q }, C(1) = {g 1 j : q (1) T q j = 1, 2}, dim(l(3)) = 1 and dim(l(1)) = 1. Daniel Tubbenhauer Consequences of cellularity - Uq -tilting view April / 26

21 An alternative semi-simplicity criterion Theorem(Graham-Lehrer 1996) Let A be a cellular algebra with cell modules C(λ) and simple modules L(λ). A is semi-simple C(λ) = L(λ) for all λ P 0. We can prove an alternative statement in our framework. Theorem The algebra End Uq (T ) is semi-simple iff T is a semi-simple U q -module. Corollary The algebra End Uq (T ) is semi-simple iff T has only simple Weyl factors. Daniel Tubbenhauer Consequences of cellularity - Uq -tilting view April / 26

22 Exempli gratia (Temperley-Lieb yet again) Because T v = v (3) v (1) v (1), and v (3) and v (1) are simple Weyl factors, we see that End Uv (sl 2)(T v ) is semi-simple. T q has a Weyl factor of the form q (3). This is a non-simple Weyl factor and thus, End Uq(sl 2)(T q ) is non semi-simple. Similarly: T L d (δ) = End Uq(sl 2)( q (1) d ) with δ 0 is semi-simple iff q is not a root of unity in K or d < ord(q 2 ). Daniel Tubbenhauer Consequences of cellularity - Uq -tilting view April / 26

23 A unified approach to cellularity - part 1 Note that our approach generalizes, for example to the infinite dimensional world: the following list is just the tip of the iceberg. The following algebras fit in our set-up as well: The Iwahori-Hecke algebra of type A, by Schur-Weyl duality: Φ qsw : H d (q) End Uq (T ) and Φ qsw : H d (q) = End Uq (T ), if n d. This includes K[S d ] for char(k) = p > 0. sl 2 -related algebras like Temperley-Lieb T L d (δ). Spider algebras End Uq(sl n)( q (ω i1 ) q (ω id )). Daniel Tubbenhauer Examples that fit into the picture April / 26

24 A unified approach to cellularity - part 2 Take g = gl m1 gl mr with m m r = m and let V be the vector representation of U 1 (gl m ) restricted to U 1 = U 1 (g). Use T = V d and Φ cl : C[Z/rZ S d ] End U1 (T ) and Φ cl : C[Z/rZ S d ] = End U1 (T ), if m d. This gives the cyclotomic analogon of the first point above. Let U q = U q (g). We get in the quantized case Φ qcl : H d,r (q) End Uq (T ) and Φ qcl : H d,r (q) = End Uq (T ), if m d, where H d,r (q) is the Ariki-Koike algebra. Daniel Tubbenhauer Examples that fit into the picture April / 26

25 A unified approach to cellularity - part 3 Let T = q (ω 1 ) d. Let g = o 2n, g = o 2n+1 or g = sp 2n (depending on δ). Φ Br : B d (δ) End U1 (T ) and Φ Br : B d (δ) = End U1 (T ), if 2n > d, where B d (δ) is the Brauer algebra in d strands. Let U q = U q (gl n ) and T = q (ω 1 ) r ( q (ω 1 ) s ) : Φ wbr : B n r,s([n]) End Uq (T ) and Φ wbr : B n r,s([n]) = End Uq (T ), if n r+s, where B n r,s([n]) the so-called quantized walled Brauer algebra. Quantizing the Brauer case: taking q K {0, ±1}, g, and T as above: the algebra End Uq (T ) is a quotient of the Birman-Murakami-Wenzl algebra BMW d (δ) and taking n d recovers BMW d (δ). Daniel Tubbenhauer Examples that fit into the picture April / 26

26 There is still much to do... Daniel Tubbenhauer Examples that fit into the picture April / 26

27 Thanks for your attention! Daniel Tubbenhauer Examples that fit into the picture April / 26