The Distinguished Involutions with a-value 3 in the Affine Weyl Group Ẽ6

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1 International Mathematical Forum, 5, 010, no. 6, The Distinguished Involutions with a-value in the Affine Weyl Group Ẽ6 Xigou Zhang College of Math. and Information on Science JXNU, Nanchang, 00, P.R. China Abstract In this paper,by using Shi s algorithm we describe the left cells in two-sided cells with a-value of the affine Weyl group Ẽ6 and constructure their graphs Fig.i(i=1,,,) of left cells. We also describe the distinguished involution in these left cells, and we can get these distinguished involutions from Fig.i. Keywords: two-sided cells, left cells, a-function, distinguished involutions 1 Preliminaries 1.1. Let W be a Coxeter group with S its generator set. Denote by the Bruhat order on W. the notation y w in W means that there exist some reduced forms w = s 1 s...s k and y = s i1 s i...s it with s i S such that i 1, i,...,i t is a subsequence of 1,,..., k. For w W, we denote by l(w) the length of w. 1..[] Let A = Z[u, u 1 ] be the ring of all the Laurent polynomials in an indeterminate u with integer coefficients. The Hecke algebra H of W over A has two A-bases { T x } x W and {C w } w W satisfying the following relations: C w = y w u l(w) l(y) P y,w (u ) T y, (1..1) where P y,w Z[u] satisfies that P w,w = 1, P y,w = 0 if y w and deg P y,w (1/)(l(w) l(y) 1) if y < w. The P y,w s are called Kazhdan-Lusztig polynomials. 1.. For y, w W with l(y) l(w), denote by µ(y, w) or µ(w, y) the coefficient of u (1/)(l(w) l(y) 1) in P y,w. The elements y and w are called jointed, written y w, if µ(y, w) 0. To any x W, we associate two subsets of S: L(x) = {s S sx < x} and R(x) = {s S xs < x}.

2 60 Xigou Zhang Moreover, {C s s S} forms a generator set of the algebra H over A. 1.. From now on, we always assume that W is an irreducible affine Weyl group or Weyl group unless otherwise specified. Define h x,y,z A by C x C y = h x,y,z C z z for any x, y, z W In [], Kazhdan and Lusztig defined The preorder L (resp. R, LR ) on elements of W. The equivalence classes of W with respect to L (resp. R, LR ) are called the left ( resp. right, two-sided) cells of W and the preorder L (resp. R, LR ) on elements of W induces a partial order on the left (resp., right, two-sided) cells of W. It is well known that (a) x L y if and only if x 1 R y 1 ; (b) x L y if and only if x 1 R y 1 ; (c)if x L y, thenx LR y; (d) if x R y, then x LR y 1.6. In [], Lusztig defined a function a : W N { } by setting a(z) = min{k N u k h x,y,z Z[u], x, y W } foranyz W The following are some known properties of the a-function: (1) If x LR y then a(x) a(y). In particular, x LR y implies a(x) = a(y). So we may define the a-value a(γ) on a left (resp. right, two-sided) cell Γ of W to be a(x) for any x Γ. () a(w J ) = l(w J ) for any J S with W J finite, where W J is the subgroup of W generated by J and w J is the longest element in W J. () For x, y, w W, we use the notation w = x y to mean that w = xy and l(w) = l(x) + l(y). In this case, we have w L y, w R x and a(w) a(x), a(y). () If a(x) = a(y) and x L y (resp. x R y) then x L y ( resp. x R y). (5) Let δ(z) = deg P e,z for z W, where e is the identity of the group W. Then the inequality l(z) δ(z) a(z) 0 holds for any z W. For i N, define D i = {w W l(w) δ(w) a(w) = i} Then Lusztig proved that D 0 is a finite set of involutions (called distinguished involution and that each left (resp. right) cell of W contains a unique element of D 0. For any x W, we have h x 1,x,d 0 for d D 0 with d L x.

3 The distinguished involutions with a-value in the affine Weyl group Ẽ6 61 Let W (i) = {w W a(w) = i} for any nonnegative integer i. Then by (1), W (i) is a union of some two-sided cells of W. (6) If W (i) contains an element of the form w I for some I S, then {w W (i) R(w) = I} forms a single left cell of W Call s S special if the group W S\{s} has the maximum possible order among all the standard parabolic subgroups of W a of the form W I, I S. For s S, let Y s = {w W a R(w) {s}}. Then Lusztig and Xi proved the following theorem []: Theorem: Let s S be special. Then Ω Y s is non-empty and forms a single left cell of W a for any two-sided cell Ω of W a. Graphs and the left cell graphs.1. In the present section, we assume that (W a, S) is an irreducible affine Weyl group of simply-laced type, that is, the order o(st) of the product st is not greater than for any s, t S, or equivalently, W a is of type Ã, D or Ẽ. Given s t in S with o(st) =, a set of the form {ys, yst} is called a (right) {s, t}-string (or a string in short), if R(y) {s, t} =. And we call that x is obtained from w by a (right) {s, t}-star operation (or a star operation in short), if {x, w} is an {s, t}-string. Note that the resulting element x for an {s, t}-star operation on w is always unique whenever it exists. Two elements x, y W a form a (right) primitive pair, if there exist two sequences of elements x 0 = x, x 1,...,x r and y 0 = y, y 1,...,y r in W a such that the following conditions are satisfied: (a) For every 1 i r, there exist some s i, t i S with o(s i t i ) = such that both {x i 1, x i } and {y i 1, y i } are {s i, t i }-strings. (b) x i y i for some (and then for all) 0 i r. (c) Either R(x) R(y) and R(y r ) R(x r ), or R(y) R(x) and R(x r ) R(y r ) hold. We get that x R y if {x, y} is a primitive pair. Similarly for left {s, t}-string and for left {s, t}-star operation... For any x W a, denote by M(x) the set of all the elements y W a such that there is a sequence of elements x = x 0, x 1,...,x r = y in W a with some r 0, where {x i 1, x i } is a string for every 1 i r. Define a graph M(x) associated to an element x W a as follows. Its vertex set is M(x); its edge set consists of all the two-element subsets in M(x) each of which forms a string; each vertex y M(x) is labelled by the set R(y)... A left cell graph associated to an element x W a, written M L (x), is by definition a graph, whose vertex set M L (x) consists of all the left cells Γ of W a with Γ M(x) ; two vertices Γ, Γ M L (x) are jointed by an edge, if there

4 6 Xigou Zhang are two elements y M(x) Γ and y M(x) Γ with {y, y } an edge of M(x), each vertex Γ of M L (x) is labeled by the set R(Γ). Clearly, for any x W, both graphs M(x) and M L (x) are connected. An algorithm for finding an l.c.r. set in a two-sided cell A subset K W a is called a representative set for the left cells (or an l.c.r set for brevity) of W a (resp. of W a in a two-sided cell Ω), if K Γ = 1 for any left cell Γ of W a (resp., of W a in Ω), where the notation X stands for the cardinality of a set X. Obviously, the set D 0 (see 1.6,()) is an l.c.r. set of W a. But it is not easy to find the whole set D 0 of W a explicitly in general since it may involve the complicated computation of Kazhdan-Lusztig polynomials. Shi constructed an algorithm for finding an l.c.r. set of W a in a two-sided cell, which is based on the following: Theorem.1 Let Ω be a two-sided cell of W a. Then a non-empty subset N Ω is an l.c.r. set of W a in Ω, if N satisfies the following conditions: (1) x L y for any x y in N; () For any y W a, if there exists some x N satisfying that y x, R(y) R(x) and a(y) = a(x), then there exists some z N with y L z. A subset P W a is called distinguished if P and x L y for any x y in P... For x, y W y x and R(y) R(x) hold if and only if one of the following cases occurs: : (1) {x, y} is a right string; () y = x s for some s S with R(y) R(x),where a = b c(a, b, c W)means that a = bc and l(a) = l(b) + l(c); () y < x, y x and R(y) R(x)... For a non-empty subset in a two-sided cell Ω of W a, consider the following processes:[shi] (A) Find a distinguished subset Q of the largest possible cardinality from the set x P M(x). (B) Let B x = {y W y 1 x S,R(y) R(x), a(y) = a(x)} for any x P. Find a distinguished subset Q of the largest possible cardinality from the set B = P ( x P B x ). (C) Let C x = {y W a y < x, y x, R(y) R(x), a(y) = a(x)} for any x P. Find a distinguished subset Q of the largest possible cardinality from the set C = P ( x P C x )... A subset P of W a is A-saturated (resp., B-saturated, C-saturated), if the

5 The distinguished involutions with a-value in the affine Weyl group Ẽ6 6 Process A (resp. B, C) on P cannot produce any element z with z x for all x P. Clearly, a set of the form x K M(x) for any K W a is always A-saturated. An l.c.r. set of W a in a two-sided cell Ω is exactly a distinguished subset of Ω which is ABC-saturated simultaneously..5. In order to get such a subset, we apply the following Algorithm[Shi s]: (1) Find a non-empty subset P of Ω (It is usual to take P distinguished and consisting of elements of the form w I, I S, whenever it is possible); () Perform Processes A, B and C alternately on P until the resulting distinguished set cannot be further enlarged by any of these processes. The left cells with a-value of Ẽ6 From now on, we let W = Ẽ6,W (i) = {x W a(x) = i},by [1], W (i) (i = 0, 1,, ) is a single two-sided cell. The left cell graphs are in [7]. For the sake of simplifying the notation, denote by i (bold-faced) the reflection s i..1. The two-sided cell W (0) consists of a single element. The two-sided cell W (1) consists of all the non-identity elements of Ẽ6 each of which has a unique reduced expression. The set S forms an l.c.r. set of W (1). The left cell graphs of W (1) are as in Fig.1... For the two-sided cell W (), we consider the set P consisting of the elements of the form w I (l(w I ) = (I S)), then P = {1,15,16,1,10,,0,5,6,0,6,5,6,05,06}. For x = 1 in P, The graph M(x) is infinite. Take a connected subgraph M L (x) from M(x). The vertex of M L (x) are distinguished, and is also A-,B-,andC--saturated. So the vertex of M L (x) forms an l.c.r. set of W (). The graph of M L (x) is as in Fig.... For W(), P = {11,,,5,0,565,16,10,150,15,16,160,5,50,6,60,60}, Since there is a single two-cell in W (). We choose x = 11, and consider the graph M(x ), The vertex of M L (x ) are A-saturated, but are not B-saturated. So we choose y = 110 from M L (x ). Let y = y5, y 1 = y0, y 1 = y 0,y = y 1, y = y 1,y = y, y = y, then R(y) = {1,0},R(y ) = {1,5,0},R(y 1 ) = {1,},R(y 1 ) = {1,,5},R(y ) = {1,},R(y ) = {1,},R(y ) = {1,},R(y ) = {}, thus {y, y } form a right primitive pair and a(y ) =. The graph M(y ) is isomorphism to the graph M(10). The sets M(y ) and M(10) represent the same set of left cells in W () since both contain a vertex labelled by 5. And the left cell graphs M L (x ) M L (10) are A-,B-,and

6 6 Xigou Zhang C-saturated. Thus The vertex of M L (x ) M L (10) forms an l.c.r. set of W (). The graphs M L (x ) and M L (10) are as in Fig. Fig.. 5 The distinguished involutions and their graphs For Weyl group and affine Weyl group, there is unique distinguised involution in every left(right) cell. In this section, we give the l.c.r set consising of the distinguised involutions with W (i) (i ) in Ẽ6.[6]. Lemma 5.1 For Weyl group and affine Weyl group, if x D 0, y = k 1 xk, where y is obtained by a right{s, t}-star operation of x, followed by similar left {s, t}-star operation for s, t S. then y D 0. Lemma 5. For Weyl group and affine Weyl group, let I S, and let w I be the longest element in the parabolic subgroupw I, then w I D The distinguished involution in W (0) is trivial. For W (1), we choose the element x 1 = 1 of form w I, by 5.1. Lemma, we get the distinguished involution graph of W (1) as in Fig.1. For W (), we choose the element x = 1 of form w I, also by 5.1.Lemma, we get the distinguished involution graph of W () as in Fig.. For W (), we choose the element x = 11 and x = 10 of form w I, by 5.1.Lemma, we get the distinguished involution graph of W () as in Fig. Fig Some notes for the distinguished involution graph: The Fig.i is corresponding to Fig.i for i = 1,,,. The Fig.i is obtained as follows: (1) The vertex of Fig.i is the distinguished involution of the corresponding left cell in the graph Fig.i. The starting elements (of course a distinguished involution) of Fig.i are listed in 5... () Every edge correspond to the edge in Fig.i is labelled by s or st. If labelled by s, then one of the vertex is obtained by the other times s from left and from right. If labelled by st, then one of the vertex is obtained by the other times s from left and times t from right. 6 Some results about the distinguished involutions Let Γ is a left cell with a in W. We define the set E min (Γ) = {w Γ l(x) l(w), x Γ} and E(Γ) = {w Γ a(sw) < a(w), s L(w)}. Clearly E min (Γ) E(Γ).

7 The distinguished involutions with a-value in the affine Weyl group Ẽ6 65 From above section, we get that for w Γ there exists y W, such that w = w J y. In order to get the set E min (Γ), we can take following steps: (i)let X 0 be the set consisting of the element of form w J, where J S and l(w J ). (ii) Set k > 0 and X k = {xs x X k 1, s S \ R(x), xs Γ}. We can get the following results from the graphs of the distinguished involutions: Theorem 6.1 Let W be the affine Weyl group Ẽ6 and Γ be a left cell with a, d L be the distinguished involution in Γ. If w E min and w = w J y. Then d L = y w J y. Theorem 6. Let W be the affine Weyl group Ẽ6 and Γ be a left cell with a, then E(Γ) = E min (Γ). ACKNOWLEDGEMENTS: The research work of the author is supported by the Natural Science Foundation of P.R.China and JiangXi Province. And it is also supported by the Science Foundation of Education Department of JiangXi Province.

8 66 Xigou Zhang Fig Fig x Fig x 5 Fig.

9 The distinguished involutions with a-value in the affine Weyl group Ẽ x Figure. Fig.

10 68 Xigou Zhang Figure x Figure.

11 The distinguished involutions with a-value in the affine Weyl group Ẽ6 69 References [1] R.W.Carter, Finite groups of Lie type:conjugacy class and complex characters, wiley Interscience, London, [] D.Kazhdan and G.Lusztig, Representations of Coxeter groups and Hecke algebras,invent. Math, 5 (1979), [] G.Lusztig, Cells in affine Weyl groups,in Alebraic Groups and Related Topics, Advanced Studies in Pure Math, [] G.Lusztig and Xi Nan-hua, Canonical left cells in affine Weyl groups, Advances in Math, 7 (1988), [5] Jian-yi Shi, Left cells in the affine Weyl group, Tohoku J.Math, 6(1) (199), [6] Jian-yi Shi, he joint relations and the set D 1 in certain crystallographic groups, Advances in Mathematics, 81 (1990), [7] Jian-yi Shi and Xi-gou Zhang, Left cells in the affine Weyl group Ẽ6, preprint. [8] J. Y. Shi and X.G.Zhang, Left cells with a-value in the affine weyl groups Ẽ i (i = 6, 7, 8),Comm. in Algebra, 6(008),17-6. [9] Xi-gou Zhang, Alcoves of non- Weyl group H, International Mathematical Forum, 1() (009), [10] Xi-gou Zhang, The elements of mini. length in the conjugacy class of the longest element in the finite Coxeter groups, JP Joural of Algebra and Applications 1(1) (009), Received: July, 009