Dual braid monoids and Koszulity

Transcription

1 Dual braid monoids and Koszulity Phillippe Nadeau (CNRS, Univ. Lyon 1) SLC 69, Strobl, September 10th 2012

2 Goal Investigate the combinatorics of a certain graded algebra associated to a Coxeter system, namely The Koszul dual of the algebra of the dual braid monoid

3 Goal Investigate the combinatorics of a certain graded algebra associated to a Coxeter system, namely The Koszul dual of the algebra of the dual braid monoid Most of the talk will be focused on type A for simplicity.... and also because I cannot yet prove the main results in all generality.

4 Usual noncrossing partitions. Let (W, S) the Coxeter system of type A n 1. So W = S n generated by S = {(i, i + 1)} for i = 1,..., n 1. Standard theory Length l S (w) = minimal k such that w = s i1 s ik. Bruhat order: w S w if l S (w) + l S (w 1 w ) = l S (w )

5 Usual noncrossing partitions. Let (W, S) the Coxeter system of type A n 1. So W = S n generated by S = {(i, i + 1)} for i = 1,..., n 1. Standard theory Length l S (w) = minimal k such that w = s i1 s ik. Bruhat order: w S w if l S (w) + l S (w 1 w ) = l S (w ) Dual presentation W with all transpositions T = {(i, j)} as generators. Absolute length l T (w) = minimal k with w = t i1 t ik. = n { cycles of w} Absolute order: w T w if l T (w) + l T (w 1 w ) = l T (w ).

6 Usual noncrossing partitions. Let (W, S) the Coxeter system of type A n 1. So W = S n generated by S = {(i, i + 1)} for i = 1,..., n 1. Standard theory Length l S (w) = minimal k such that w = s i1 s ik. Bruhat order: w S w if l S (w) + l S (w 1 w ) = l S (w ) Dual presentation W with all transpositions T = {(i, j)} as generators. Absolute length l T (w) = minimal k with w = t i1 t ik. = n { cycles of w} Absolute order: w T w if l T (w) + l T (w 1 w ) = l T (w ). NC(n) = NC(A n 1 ) = [id, (12 n)] T

7 NC(4) and its Möbius function (1234) (123)(4) (14)(23) (124)(3) (134)(2) (12)(34) (1)(234) (12)(3)(4) (13)(2)(4) (1)(23)(4) (14)(2)(3) (1)(24)(3) (1)(2)(34) (1)(2)(3)(4)

8 NC(4) and its Möbius function µ(w) = µ T (id, w)

9 NC(4) and its Möbius function µ(w) = µ T (id, w) 5 ( 1) n 1 C n

10 Braids Braids on n strings = Strings go down, 2 of them can cross in two ways. Braids (up to isotopy) form a group wrt concatenation.

11 Braids Braids on n strings = Strings go down, 2 of them can cross in two ways. Braids (up to isotopy) form a group wrt concatenation. Permutation of the endpoints S n as quotient group.

12 Braids Consider the monoid BKL n generated by the braids a ij in this braid group. (Usual generators of the braid group/monoid are a ii+1.) Braids on n strings = Strings go down, 2 of them can cross in two ways. Braids (up to isotopy) form a group wrt concatenation. Permutation of the endpoints S n as quotient group. i a ij j

13 Braids a ij a jk a jk a ik Consider the monoid BKL n generated by the braids a ij in this braid group. (Usual generators of the braid group/monoid are a ii+1.) Braids on n strings = Strings go down, 2 of them can cross in two ways. Braids (up to isotopy) form a group wrt concatenation. Permutation of the endpoints S n as quotient group. i a ij j They verify certain relations, eg a ij a jk = a jk a ik. One can characterize all such relations.

14 The Birman Ko Lee monoid Proposition [BKL 98] The monoid BKL n has generators a ij for 1 i < j n and relations: a ij a jk = a jk a ik = a ik a ij for i < j < k; a ij a kl = a kl a ij for i < j < k < l or i < k < l < j. The relations respect length of words length l(m) of an element in the quotient is well defined.

15 The Birman Ko Lee monoid Proposition [BKL 98] The monoid BKL n has generators a ij for 1 i < j n and relations: a ij a jk = a jk a ik = a ik a ij for i < j < k; a ij a kl = a kl a ij for i < j < k < l or i < k < l < j. The relations respect length of words length l(m) of an element in the quotient is well defined. Let P n (t) = Proposition For instance n 1 k=0 (n 1 + k)! (n 1 k)!k!(k + 1)! tk. m BKL n t l(m) = m BKL 4 t l(m) = 1 P n ( t) t + 10t 2 5t 3.

16 Proof of the evaluation of P n (t) In fact there is a well known relation between the length generating function and the Moebius function of the monoid ordered by divisibility (see [Cartier Foata]). P n (t) = µ(m) t l(m). m BKL n

17 Proof of the evaluation of P n (t) In fact there is a well known relation between the length generating function and the Moebius function of the monoid ordered by divisibility (see [Cartier Foata]). P n (t) = µ(m) t l(m). m BKL n But BKL n is a Garside monoid with Garside element C = a 12 a 23 a n 1n µ(m) = 0 if m does not divide C. Furthermore [1, C] left divisibility NC(n), and so P n (t) = w NC(S n ) µ(w) tl(w)

18 Proof of the evaluation of P n (t) In fact there is a well known relation between the length generating function and the Moebius function of the monoid ordered by divisibility (see [Cartier Foata]). P n (t) = µ(m) t l(m). m BKL n But BKL n is a Garside monoid with Garside element C = a 12 a 23 a n 1n µ(m) = 0 if m does not divide C. Furthermore [1, C] left divisibility NC(n), and so P n (t) = w NC(S n ) µ(w) tl(w) In [Albenque, N. 09] we computed this combinatorially.

19 Proof of the evaluation of P n (t) In fact there is a well known relation between the length generating function and the Moebius function of the monoid ordered by divisibility (see [Cartier Foata]). P n (t) = µ(m) t l(m). m BKL n But BKL n is a Garside monoid with Garside element C = a 12 a 23 a n 1n µ(m) = 0 if m does not divide C. Furthermore [1, C] left divisibility NC(n), and so P n (t) = w NC(S n ) µ(w) tl(w) In [Albenque, N. 09] we computed this combinatorially. And then I talked to Vic Reiner.

20 The monoid algebra We pass from the monoid to its k-algebra A: Definition The algebra A is defined by the generators a ij and relations I =<a ij a kl a kl a ij for i < j < k < l or i < k < l < j; a ij a jk a jk a ik ; a jk a ik a ik a ij for i < j < k.

21 The monoid algebra We pass from the monoid to its k-algebra A: Definition The algebra A is defined by the generators a ij and relations I =<a ij a kl a kl a ij for i < j < k < l or i < k < l < j; a ij a jk a jk a ik ; a jk a ik a ik a ij for i < j < k. Elements of BKL n form a basis of A, which has a grading A = k 0 A k, with Hilb A (t) = n 0 dim A k t k = 1 P n ( t) We associate to A another algebra A, the Koszul dual of A. This transformation Q Q is defined more generally for all quadratic algebras Q.

22 Koszul duality of quadratic algebras Definition A quadratic algebra Q is a graded algebra where the ideal of relations is generated by elements of degree 2. Q = k x 1,..., x m /Ideal(R) with R vector subspace of W 2 := { i,j λ ijx i x j }.

23 Koszul duality of quadratic algebras Definition A quadratic algebra Q is a graded algebra where the ideal of relations is generated by elements of degree 2. Q = k x 1,..., x m /Ideal(R) with R vector subspace of W 2 := { i,j λ ijx i x j }. Declare that (x i x j ) i,j is an orthonormal basis of W 2, and let R be the orthogonal of R. Then define Q = k x 1,..., x m /Ideal(R )

24 Koszul duality of quadratic algebras Definition A quadratic algebra Q is a graded algebra where the ideal of relations is generated by elements of degree 2. Q = k x 1,..., x m /Ideal(R) with R vector subspace of W 2 := { i,j λ ijx i x j }. Declare that (x i x j ) i,j is an orthonormal basis of W 2, and let R be the orthogonal of R. Then define Examples (a) R = span{x i x j x j x i, i < j} Symmetric algebra Q = k x 1,..., x m /Ideal(R ) R =span{x 2 i, x ix j + x j x i, i < j} Exterior algebra (b) R = span{x i x j, (i, j) I [m] 2 } R = span{x i x j, (i, j) [m] 2 I} Monomial ideals

25 Koszul duality for algebras We have A = k a ij /Ideal(R) with R = span{a ij a kl a kl a ij for i < j < k < l or i < k < l < j a ij a jk a jk a ik ; a jk a ik a ik a ij for i < j < k}.

26 Koszul duality for algebras We have A = k a ij /Ideal(R) with R = span{a ij a kl a kl a ij for i < j < k < l or i < k < l < j a ij a jk a jk a ik ; a jk a ik a ik a ij for i < j < k}. What is R in this case? It has a basis consisting of: (1) all a i,j a k,l which do not appear above; (2) a i,j a k,l + a k,l a i,j for (i, j), (k, l) noncrossing; (3) a ij a jk + a jk a ik + a ik a ij with i < j < k.

27 Koszul duality for algebras We have A = k a ij /Ideal(R) with R = span{a ij a kl a kl a ij for i < j < k < l or i < k < l < j a ij a jk a jk a ik ; a jk a ik a ik a ij for i < j < k}. What is R in this case? It has a basis consisting of: (1) all a i,j a k,l which do not appear above; (2) a i,j a k,l + a k,l a i,j for (i, j), (k, l) noncrossing; (3) a ij a jk + a jk a ik + a ik a ij with i < j < k. Theorem [Albenque, N. 09; N. 12] Hilb A (t) = P n (t)

28 Koszul duality for algebras We have A = k a ij /Ideal(R) with R = span{a ij a kl a kl a ij for i < j < k < l or i < k < l < j a ij a jk a jk a ik ; a jk a ik a ik a ij for i < j < k}. What is R in this case? It has a basis consisting of: (1) all a i,j a k,l which do not appear above; (2) a i,j a k,l + a k,l a i,j for (i, j), (k, l) noncrossing; (3) a ij a jk + a jk a ik + a ik a ij with i < j < k. Theorem [Albenque, N. 09; N. 12] Hilb A (t) = P n (t) Main question: why did I reprove one of my own results?

29 Koszul algebras In [Albenque, N. 09], we proved a bit more. We showed that A is a Koszul algebra, which can be defined as A graded k-algebra Q such that the Q-module k admits a minimal graded free resolution which is linear.

30 Koszul algebras In [Albenque, N. 09], we proved a bit more. We showed that A is a Koszul algebra, which can be defined as A graded k-algebra Q such that the Q-module k admits a minimal graded free resolution which is linear. Now Q Koszul Q numerically Koszul: Hilb Q (t) Hilb Q ( t) = 1 But this gives no insight on the structure of A.

31 Koszul algebras In [Albenque, N. 09], we proved a bit more. We showed that A is a Koszul algebra, which can be defined as A graded k-algebra Q such that the Q-module k admits a minimal graded free resolution which is linear. Now Q Koszul Q numerically Koszul: Hilb Q (t) Hilb Q ( t) = 1 But this gives no insight on the structure of A. New work: a nice basis of the algebra A. A = A [w] w N C(n) with an explicit basis of A [w] of cardinality µ(w).

32 Other Coxeter groups (W, S) a finite Coxeter system Reflections t T = w wsw 1 as new generators.

33 Other Coxeter groups (W, S) a finite Coxeter system Reflections t T = w wsw 1 as new generators. l T (w) = minimal k such that w = t 1 t k. w T w if l T (w) + l T (w 1 w ) = l T (w ) Fix a Coxeter element c, define NC(W ) := [1, c] T.

34 Other Coxeter groups (W, S) a finite Coxeter system Reflections t T = w wsw 1 as new generators. l T (w) = minimal k such that w = t 1 t k. w T w if l T (w) + l T (w 1 w ) = l T (w ) Fix a Coxeter element c, define NC(W ) := [1, c] T. [Bessis 00] Define the dual braid monoid as generated by a t with t T and relations a t a u = a u a utu whenever tu T c.

35 Other Coxeter groups (W, S) a finite Coxeter system Reflections t T = w wsw 1 as new generators. l T (w) = minimal k such that w = t 1 t k. w T w if l T (w) + l T (w 1 w ) = l T (w ) Fix a Coxeter element c, define NC(W ) := [1, c] T. [Bessis 00] Define the dual braid monoid as generated by a t with t T and relations a t a u = a u a utu whenever tu T c. Its algebra A(W ) is clearly quadratic, we can therefore consider the dual algebra A (W ) and explicit a presentation. Same questions for A(W ) instead of A = A(S n )

36 Questions for the future 1) Is A(W ) numerically Koszul, ie. Hilb A(W ) (t) Hilb A (W )( t) = 1? 2) Is A(W ) a Koszul algebra? 3) Is there a decomposition A (W ) = w NC(W ) A (W )[w] with A (W )[w] has a nice basis of size µ(w)?

37 Questions for the future 1) Is A(W ) numerically Koszul, ie. Hilb A(W ) (t) Hilb A (W )( t) = 1? 2) Is A(W ) a Koszul algebra? 3) Is there a decomposition A (W ) = w NC(W ) A (W )[w] with A (W )[w] has a nice basis of size µ(w)? Conjecture Yes.

38 Questions for the future 1) Is A(W ) numerically Koszul, ie. Hilb A(W ) (t) Hilb A (W )( t) = 1? 2) Is A(W ) a Koszul algebra? 3) Is there a decomposition A (W ) = w NC(W ) A (W )[w] with A (W )[w] has a nice basis of size µ(w)? Conjecture Yes. Known and To do 3) or 2) imply 1). 2) is true for type B [Albenque, N. 09]. Check 3) (or simply 1) for exceptional types by computer. Prove 3) by checking that a certain chain complex is exact (V. Féray). Use explicit EL-shelling of NC(W).