D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165-184.  Home/Search   Document Not in Database   Summary   Related Articles   Check

..... What is interesting (Theorem 4.2.5) is that there are important cases of this form where B is precisely the canonical basis (in the sense of [8] of a certain quotient of a Hecke algebra of type A, B, H or I. We conclude that there is a close relationship between the Kazhdan Lusztig bases of [13] on the one hand and certain wreath products of discrete hypergroups with Temperley Lieb algebras on the other. Although our main results consider the case where the hypergroups of the title are Verlinde algebras, we develop the theory more generally because it is useful in other contexts, such ....

....The set ft w : w 2 W c g is an A basis for the algebra TL(X) Proof. See [4, Theorem 6. 2] We now recall a principal result of [8] which establishes the canonical basis for TL(X) This basis is a direct analogue of the important Kazhdan Lusztig basis of the Hecke algebra H(X) defined in [13]. Fix a Coxeter graph, X. Let A = Z[v ] and let be the involution on the ring A = Z[v; v ] which satisfies v = v . By [8, Lemma 1.4] the algebra TL(X) has a Z linear automorphism of order 2 that sends v to v and t w to t Gamma1 . We denote this map also by . Let L be the ....

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D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165--184.

....(ii) is also interesting in that it captures the notion of 321 avoiding by considering the images of only two elements. 3. Kazhdan Lusztig cells We wish to apply Theorem 2.7 in order to understand better the Kazhdan Lusztig cells in type e A n 1 . For any Coxeter group W , Kazhdan and Lusztig [13] de ned partitions of W into left cells, right cells and two sided cells, where the two sided cells are unions of left (or right) cells. These cells are naturally partially ordered by an explicitly de ned order, LR . We do not present the full de nitions here; an elementary introduction to the ....

.... s 2k , the partition (w) has k parts equal to 2 and the other parts equal to 1. The result follows. We end by discussing brie y the application of Theorem 3.4 to the computation of certain structure constants for the Kazhdan Lusztig basis. The Kazhdan Lusztig basis, which rst appeared in [13], is a free Z[v; v ] basis for the Hecke algebra H(W ) associated to the group W . The basis, fC w : w 2 Wg, is naturally indexed by W . The structure constants g x;y;z , namely the Laurent polynomials occurring in the expression C y = z2W g x;y;z C z ; have many subtle ....

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

....The matrix coefficients of # relative to the E mail address: springer math.uu.nl. 0021 8693 02 see front matter 2002 Elsevier Science (USA) All rights reserved. PII: S0021 8693(02)00515 X basis ) are discussed at the end of Section 3. They bear some resemblance to the R polynomials of [KL]. In Section 4 it is shown that the intersection cohomology of an orbit closure leads to Kazhdan Lusztig elements in M. The results about matrix coefficients of Section 3 together with results of [MS] imply the evenness of local intersection cohomology, and the existence of Kazhdan Lusztig ....

....if d(t .w) d(w) 10) Using these formulas, by induction is a polynomial in u 1 u, hence is invariant under the change u ## u 1 . Then (ii) follows from (i) It remains to deal with the casev 0,D w 0,J , 1] In the sequel the R polynomials of Kazhdan Lusztig (see [KL, Section 2]) will appear. They lie in Z[u ] They are defined in terms of the Hecke algebra x 1 = u 2l(x) e y , W. From (7) we deduce [D,1,y] D,1,x] R y,x . We have R y,x 0ify x and R x,x 1. The R polynomials satisfy the following recursive relations (where x,y W , s ....

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D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165--184.

....basis (IC basis) Formally, this is similar to the basis fC w : w 2 Wg for the Hecke algebra introduced by 2000 Mathematics Subject Classification. 20F55, 20C08, 57M15. The author was supported in part by an award from the Nuffield Foundation. Typeset by A M S T E X Kazhdan and Lusztig [10], although the precise relationship between the two is not completely obvious. The goal of this paper is to tie these two theories together by showing how decorated tangles may be used to describe certain canonical bases for generalized Temperley Lieb algebras. This is easily done in types D ....

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165--184.

....that e w = e w Gamma1 . One shows that for w 2 W there is a unique element c w 2 H with c w = c w , of the form c w = t )e x ; where, the P x;w being polynomials with Pw;w = 1, 2deg P x;w l(w) Gamma l(x) if x w. The c w form a basis of H, the Kazhdan Lusztig basis (introduced in [KL1], see also [Hu2, II, 7] We shall now connect the Kazhdan Lusztig elements with the intersection cohomology complexes I w = IC(Ow ; Q) of the G orbit closures in X Theta X . For s 2 S we have I s = A s [1] where A s is as in 2.4. Also, h is as in 2.4. Theorem. Let x; w 2 W . i) h(I w ....

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Inv. Math. 53 (1979), 165-184.

....are in canonical correspondence with elements of H(W ) acting by right multiplication. 2.4 The Kazhdan Lusztig basis. Using the methods of Du [7] we can describe a second basis for the algebra b S q (n; r) analogous to the Kazhdan Lusztig basis for the Hecke algebra of a Coxeter group [19]. Such a basis is called an IC (intersection cohomology) basis; this terminology is justified in the case of the ordinary q Schur algebra, where the basis has intimate connections to Lusztig s canonical basis for U (the plus part of a quantized enveloping algebra) and the theory of perverse ....

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165--184.

....4 we give a canonical embedding of each cell representation as a submodule of H . 1 Left cells and Lusztig s a function In this section we set our notation and recall the results from the literature which we shall need. Most of these preliminary results are due to either Kazhdan and Lusztig [5, 6] or Lusztig [10, 13] a good reference is Curtis survey article [1] For brevity we assume throughout that W is a finite Weyl group. The ring A has two semi linear involutions and given by q and (q ) Gammaq . Extend these to involutions on H by setting T x = T Gamma1 and (T ....

....2 H . For convenience we renormalise the T basis of H and write T x = q x T x for all x 2 W . Then ( T x and ( T x T y ) ffi x;y Gamma1 for all x; y 2 W (Kronecker delta) Let denote the Bruhat ordering on W and recall the Kazhdan Lusztig bases fC x g and fC x g of H from [5] (x 2 W ) These elements are characterized as the unique elements of H such that (i) C x = C x and C x = C x and T y and C x = ffl x q T y where P y;x 2 Z[q] is a polynomial such that P x;x = 1 for all x and if y x then P y;x has degree (in q) of at most 2 ( x) ....

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D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165--184.

....Kac Moody case are similar to the finite case, and hence we have been brief and outlined only the necessary changes. There are other criteria for smoothness due to Lakshmibai Seshadri (for classical groups) LS] L] Ryan (for SL(n) Ry] and for rational smoothness due to Kazhdan Lusztig [KL], Carrell Peterson [C] Jantzen [J] and by works of Deodhar and Peterson rational smoothness implies smoothness for simply laced groups. It may be mentioned that our criterion for smoothness (as in Theorem 5.5(b) is applicable to all G uniformly, in contrast to the above mentioned ....

.... Gamma g : 5) This proves the proposition. 5.3) Remark. When the equivalent condition as in the above Proposition (5. 2) is satisfied, d in fact is an integer 0 (as is clear from the above proof) We recall the definition of a rationally smooth point in a variety Y (cf.[KL, Appendix]) 5.4) Definition. A variety Y of dim d is said to be rationally smooth if for all y 2 Y , the singular cohomology H = 0 if i 6= 2d and H 2d is one dimensional. A point y 0 2 Y is said to be rationally smooth if there exists an open (in the Zariski topology) rationally smooth ....

Kazhdan, D., and Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165--184.

....B i2 ) EndA (B i1 ) Phi EndA (B i2 ) contradicting the fact that B i is an indecomposable block ideal of A. 1. All of the results in this Chapter are due to Graham and Lehrer and can be found in [25,26] 2. The introduction of cellular algebras was largely motivated by the Kazhdan Lusztig [46] bases of the Iwahori Hecke algebras of type A. Many more examples of cellular algebras can be found in [25, 26] For example, generalized) Temperley Lieb algebras, Jones annular algebra, Ariki Koike algebras, and the cyclotomic q Schur algebras of [17] are all cellular. 3. A ring theoretic ....

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165--184.

....of the group, called a descent representation. Descent representations of Weyl groups were first introduced by Solomon [31] as alternating sums of permutation representations. This concept was extended to arbitrary Coxeter groups, using a different construction, by Kazhdan and Lusztig [22] [21, x7.15] For Weyl groups of type A, these representations also appear in the top homology of certain (Cohen Macaulay) rank selected posets [34] Another description (for type A) is by means of zig zag diagrams [19] 16] In this paper we give a new construction of descent representations for ....

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

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D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

....for reducing words in small Coxeter groups. To illustrate, we give the complete code for reducing words to normal form in the Coxeter group (W, S ) of type H 3 . We define the normal form of a group element to be the lexicographically first reduced word that represents it. s : s: usage : s[1,2] is the product of reflections 1 and 2 reduce[s[a , c , c , b ] reduce[s[a, b] The next six rules are generated by the program ) reduce[s[a , 3, 1, b ] reduce[s[a, 1, 3, b] reduce[s[a , 3, 2, 3, b ] reduce[s[a, 2, 3, 2, b] reduce[s[a , 3, 2, 1, 3, b ] ....

....W I , i.e. the set of x W with xs x for all s I. Observe that we could write #(xs) #(x) instead of xs x. In other words, the length function su#ces and the partial order on W is not needed to compute W . A similar remark applies to the following description of the R polynomials of [1]. Our # is always a Kronecker #. The R polynomials satisfy the following recursive relations (where x, y W, s S ) Together with the boundary conditions R y,1 = # y,1 these relations define the R polynomials uniquely. R y,sx = R sy,x , if sx x, sy y; u 1)R y,x u R sy,x ....

D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979), 165--184.

....by a polynomial (which depends only on k) in l(v) Gamma l(u) In particular, this implies the validity of Lascoux Schutzenberger s conjecture for all sufficiently long intervals, and gives supporting evidence in favor of the Dyer Lusztig conjecture. 1 Introduction In their fundamental paper [14] Kazhdan and Lusztig defined, for every Coxeter group W , a family of polynomials, indexed by pairs of elements of W , which have become known as the Kazhdan Lusztig polynomials of W (see, e.g. 13] Chap. 7) These polynomials are intimately related to the Bruhat order of W and to the algebraic ....

....inverse Kazhdan Lusztig) polynomial of any Coxeter group and to study some consequences of these bounds. Our motivation for doing this comes from two conjectures of Kazhdan Lusztig and Lascoux Schutzenberger which assert, respectively, that these coefficients are always nonnegative (see, e.g. [14], p. 166) and that, if the polynomials have the maximum possible degree, then they are bounded from above by appropriate Eulerian numbers (see, 17] p. 249, or x2 for the precise statement of this conjecture) Our main result is that, if Partially supported by EC grant No. CHRX CT93 0400 the ....

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D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165-184. 19

....1.1. Let (W; S) be a Coxeter system, where we assume that the cardinality jSj = n of S is nite; usually we omit S from the notation and simply speak of the Coxeter group W . We will denote the Bruhat ordering on W (see [8] for its de nition and basic properties) In their celebrated paper [9], Kazhdan and Lusztig have de ned for each x y in W a polynomial P x;y 2 Z[q] which are the components of a remarkable basis of the Hecke algebra of W . When W is the Weyl group of a nite or Ka c Moody Lie algebra g, these polynomials hold the key to the representation theories of g and ....

....on Q. Proof. We prove that P (x) y) P x;y for all x y in Q by induction on l(y) and for xed y, by induction on l(y) l(x) If x = y there is nothing to prove. Assume x y, and choose s 2 S such that ys s. Then we have (zs) z)s for all z 2 [e; y] From the recursion formul in [9], 2.2.c) and (2.3.g) we have three cases : a) xs x. Then P x;y = P xs;y . b) xs x, x 6 ys. Then P x;y = P xs;ys . c) xs x ys. Then we have the recursion formula : P x;y = P xs;ys qP x;ys X x z ys zs z q 2 (l(y) l(z) z; ys)P x;z where (x; y) is the coecient of ....

D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math., 53:165-184, 1979.

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D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165-184.

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D. Kazhdan, G. Lusztig, Representation of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

....Coxeter groups. Examples of Coxeter groups are Weyl groups of nite dimensional or Ka c Moody semisimple Lie algebras, and nite groups generated by re ections in Euclidian space; other examples may be realized as discrete groups generated by re ections in hyperbolic space. In their seminal paper [15], Kazhdan and Lusztig have de ned for each pair of elements (x; y) in W such that x y in the Bruhat ordering (to be de ned below) a polynomial P x;y 2 Z[q] We will recall in section 4 the recursion formula which in principle leads to the computation of P x;y . If W is the Weyl group of a ....

....exactly two elements not already in [e; y] viz. zs and ys; in particular, coat(ys) fy; zsg, which from the Theorem implies that [e; ys] is dihedral; but then [e; y] is dihedral as well, contradicting our assumption on y. 4 Kazhdan Lusztig polynomials 4.1. We refer to the original paper [15] or to [14] chapter 7, for the proper de nition and proofs of the basic properties of the Kazhdan Lusztig polynomials. Our goal here is to recall the recursion formul from [15] which we use to compute these polynomials, in order to assess the data which will be needed in the process. Let q be ....

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D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math., 53:165-184, 1979.

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D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165-184.

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D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

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D. Kazhdan, and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165--184.

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D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165--184.

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D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

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D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

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D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Inv. Math. 53 (1979), 165--184.

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D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

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