A NEW ALGORITHM FOR FINDING AN LCR SET

A NEW ALGORITHM FOR FINDING AN L.C.R. SET IN CERTAIN TWO-SIDED CELLS

Jian-yi Shi Department of Mathematics East China Normal University Shanghai, 200241, P.R.China Abstract. Let (W, S) be an irreducible Weyl or affine Weyl group. In [18], we constructed an algorithm for finding a representative set of left cells (or an l.c.r. set for short) of W in a two-sided cell Ω. In the present paper, we introduce a new simpler algorithm for finding an l.c.r. set of W in Ω when the subset F (Ω) of Ω is known (see Algorithm 3.11). Some technical tricks are introduced by some examples for applying the algorithm and for finding the set F (Ω). The resulting set E(Ω) of Algorithm 3.4 is useful in verifying a conjecture of Lusztig that any left cell in an affine Weyl group is left-connected.

Let W be an irreducible Weyl or affine Weyl group with S its Coxeter generator set. For a two-sided cell Ω of W (in the sense of Kazhdan–Lusztig, see [7]), we introduced an algorithm for finding an l.c.r. set of W in Ω in [18]. The algorithm has been efficiently applied in many cases (see [3], [4], [13], [18], [19], [22], [23], [26], [27], [28], [29], etc) since then. The algorithm consists of three processes (A), (B), (C) on a distinguished set F (see 3.1), where process (C) is the most difficult part among the three in which one need to find, for any given x ∈ F , all elements y satisfying y—–x, y < x, R(y) * R(x) and a(y) = a(x) (see 1.1. and 1.3. for the notation). It becomes extremely difficult as the length of x is getting larger. For any two-sided cell Ω of W , let F (Ω) be the set of all w ∈ Ω such that a(sw), a(wt) < a(w) for any s ∈ L(w) and t ∈ R(w). We shall introduce a new Key words and phrases. affine Weyl groups, left cells, two-sided cells, alcove forms, algorithm. Supported by the NSF of China, the SFUDP of China, PCSIRT, Shanghai Leading Academic Discipline Project (B407) and Program of Shanghai Subject Chief Scientist (11xd1402200) Typeset by AMS-TEX

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Jian-yi Shi

algorithm for finding an l.c.r. set of W in a two-sided cell Ω, provided that the subset F (Ω) of Ω is known (see 3.2). The processes in our new algorithm amount to the mixture of processes (A) and (B) in the original algorithm, hence avoiding process (C). Theorem 3.5 guarantees that our new algorithm will terminate after a finite number of steps, while Theorem 3.12 shows that the resulting set E0 (Ω) of Algorithm 3.11 forms an l.c.r. set of W in Ω each element of E0 (Ω) is shortest in the left cell of W containing it. Our new algorithm has been applied successfully for the description of the left ei , i = 6, 7, 8 (see [6], [8], [24]). cells of a-values 4, 5, 6 in the affine Weyl groups E It is desirable to find the subset F (Ω) explicitly for more two-sided cells Ω in an irreducible Weyl and affine Weyl group in order to apply our new algorithm. It is relatively easier to describe the set F (Ω) when F (Ω) consists of elements of the form wJ for some J ⊆ S, where the subgroup WJ of W generated by J is finite and wJ is the longest element in WJ (see 4.6). We can also find the sets F (Ω) for some two-sided cells Ω when Ω contains some elements not of the form wJ , J ⊆ S (see 4.1 and 4.7 –4.10). Some technical tricks are needed in applying Algorithm 3.11. We explain by some examples the way to find the set E0 (Ω) from F0 (Ω), the set F0 (Ω) from F (Ω), the set F (Ω) from F (W(i) ) (see 1.3 and 4.7) with a(Ω) = i, and the set F (W(i) ) from some known conditions (see Section 4). Lusztig conjectured in [1] that any left cell of an affine Weyl group W is leftconnected. For a two-sided cell Ω of W , the resulting set E(Ω) of Algorithm 3.4 is useful in the verification of left-connectedness for a left cell of W (see 3.6-3.7). The contents of the paper are organized as follows. We collect some results on cells of affine Weyl groups W in Section 1 and on the alcove form of elements of W in Section 2. The most results stated in those sections are known already except for Proposition 2.3. Then we introduce a new algorithm for finding an l.c.r. set of

A new algorithm for finding an l.c.r. set

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W in a two-sided cell Ω of W in Section 3. Finally, in Section 4, we explain some technical tricks in applying the algorithm. §1. Some results on cells of affine Weyl groups. In this section, we collect some known results on cells of affine Weyl groups. 1.1. Let W = (W, S) be a Coxeter group with S its Coxeter generator set. Let 6 be the Bruhat-Chevalley order on W . For w ∈ W, we denote by `(w) the length of w. Let A = Z[u, u−1 ] be the ring of Laurent polynomials in an indeterminate u with integer coefficients. Let H(W ) be the associated Hecke algebra of W , i.e., an associative A-algebra which is free as an A-module with a basis {Tw | w ∈ W }, subject to the multiplication rule:

Tx Ty = Txy ,

if `(x) + `(y) = `(xy);

(Ts − u−1 )(Ts + u) = 0

for any s ∈ S.

H(W ) has another A-basis {Cw | w ∈ W } given by

(1.1.1)

Cw =

X

u`(w)−`(y) Py,w (u−2 )Ty ,

y6w

where the Py,w ∈ Z[u] (y, w ∈ W ) are the celebrated Kazhdan-Lusztig polynomials satisfying: Pw,w = 1, Py,w = 0 if y w, and deg Py,w 6 (1/2)(`(w) − `(y) − 1) if y < w (see [7]). For y < w in W , let µ(w, y) = µ(y, w) be the coefficient of u(1/2)(`(w)−`(y)−1) in Py,w . We denote y—–w if µ(y, w) 6= 0. Checking the relation y—–w for y, w ∈ W usually involves very complicated computation of Kazhdan-Lusztig polynomials. But it is easier in some special case: (1.1.2)

If x, y ∈ W satisfy y < x and `(y) = `(x) − 1, then y—–x.

1.2. The preorders 6, 6, 6 and the associated equivalence relations ∼, ∼, L

R

LR

L

R

∼ on W are defined as in [7]. The equivalence classes of W with respect to ∼

LR

L

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Jian-yi Shi

(respectively, ∼, ∼ ) are called left cells (respectively, right cells, two-sided cells). R

LR

The preorder 6 (respectively, 6, 6 ) induces a partial order on the set of left cells L

R

LR

(respectively, right cells, two-sided cells) of W . From now on, we always assume W an irreducible Weyl or affine Weyl group unless otherwise specified. 1.3. In [9], Lusztig defined a function a: W −→ N which satisfies the following properties: (1) If x 6 y then a(x) > a(y). In particular, if x ∼ y then a(x) = a(y). So we LR

LR

may define the a-value a(Γ) to be a(x) for any x ∈ Γ, where Γ is a left, right or two-sided cell of W (see [9]). (2) If a(x) = a(y) and x 6 y (respectively, x 6 y ) then x ∼ y (respectively, x ∼ y L

R

L

R

) (see [10]). (3) For any I ⊆ S with |WI | < ∞, we have a(wI ) = `(wI ). For any w ∈ W , set

L(w) = {s ∈ S | sw < w}

and

R(w) = {s ∈ S | ws < w}.

(4) If x 6 y (respectively, x 6 y), then R(x) ⊇ R(y) (respectively, L(x) ⊇ L(y)). L

R

In particular, if x ∼ y (respectively, x ∼ y), then R(x) = R(y) (respectively, L

R

L(x) = L(y)) (see [7, Proposition 2.4]). By the notation x = y · z (x, y, z ∈ W ), we mean x = yz and `(x) = `(y) + `(z). In this case, we call x a left (respectively, right) extension of z (respectively, y), and call z (respectively, y) a left (respectively, right) retraction of x. When w = x · y · z, we call w an extension of y and call y a retraction of w. (5) If x = y · z then x 6 z and x 6 y. Hence a(x) > a(y), a(z) by (2). In particular, L

R

if I ∈ {R(x), L(x)}, then a(x) > `(wI ) (see [9]). Let W(i) = {w ∈ W | a(w) = i} for any i ∈ N. Then by (2), W(i) is a union of some two-sided cells of W . (6) For any x ∈ W , let Σ(x) be the set of all left cells Γ of W satisfying that there

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exists some y ∈ Γ with y—x, R(y) * R(x) and a(y) = a(x). Then x ∼ y in W L

if and only if R(x) = R(y) and Σ(x) = Σ(y) (see [18, Theorem 2.1], [21] and [22, Section 5]). (7) If x—–y in W and s ∈ S satisfy s ∈ L(y) \ L(x) (respectively, s ∈ R(y) \ R(x)) then either y = s · x (respectively, y = x · s) or y < x (see [7, Subsections 2.3e and 2.3f]). In particular, we have `(y) 6 `(x) + 1. (8) The number of left cells in W is finite (see [10, Theorem 2.2]). (9) If a left cell L and a right cell R are in the same two-sided cell of W , then L ∩ R 6= ∅ (see [11, Subsection 3.1 (k), (l)]). 1.4. To each x ∈ W , we denote by M (x) the set of all y ∈ W such that there is a sequence x0 = x, x1 , ..., xr = y in W with some r > 0, where for every 1 6 i 6 r, +

the conditions x−1 i−1 xi ∈ S and R(xi−1 )* R(xi ) are satisfied. A graph M(x) associated to an element x ∈ W is defined as follows. Its vertex set is M (x), each y ∈ M (x) is labelled by the set R(y); its edge set consists of all +

two-elements subsets {y, z} ⊂ M (x) with y −1 z ∈ S and R(y)* R(z). By a path in the graph M(x), we mean a sequence z0 , z1 , . . . , zr in M (x) such that {zi−1 , zi } is an edge of M(x) for any 1 6 i 6 r. Two elements x, x0 ∈ W have the same right generalized τ -invariants, if for any path z0 = x, z1 , ..., zr in M(x), there is a path z00 = x0 , z10 , ..., zr0 in M(x0 ) with R(zi0 ) = R(zi ) for any 0 6 i 6 r, and if the same condition holds when the roles of x and x0 are interchanged. Then the following result is known. Proposition 1.5. (see [17, Section 3]) Any x, y ∈ W with x ∼ y have the same L

right generalized τ -invariants. 1.6. For s, t ∈ S with m = o(st) > 2, each of the sequences

zt, zts, ztst, ...

and

zs, zst, zsts, ...

(each contains m − 1 terms)

is called a right {s, t}-strings, where z ∈ W satisfies R(z) ∩ {s, t} = ∅.

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Jian-yi Shi Two elements x, y ∈ W form a right primitive pair, if there exist two sequences

x0 = x, x1 , . . . , xr and y0 = y, y1 , . . . , yr in W satisfying that (a) For each 1 6 i 6 r, there exist some si , ti ∈ S with o(si ti ) > 2 such that any of the pairs xi−1 , xi and yi−1 , yi are neighboring terms in a right {si , ti }-string. (b) xi —–yi for all 0 6 i 6 r. (c) Either R(x) * R(y) and R(yr ) * R(xr ), or R(y) * R(x) and R(xr ) * R(yr ). In particular, if {w, y} is an edge in a graph M(x) for some x ∈ W then w, y form a right primitive pair by taking r = 0 in the above definition. Similarly, we can define a left {s, t} string and a left primitive pair. Proposition 1.7. (see [17, Section 3]) Any right (respectively, left) primitive pair x, y ∈ W satisfy x ∼ y (respectively, x ∼ y). In particular, the set M (x) is R

L

contained in a right cell of W . 1.8. An affine Weyl group W is a Coxeter group which can be realized geometrically as follows. Let G be a connected, adjoint reductive algebraic group over C. Fix a maximal torus T of G. Let X be the group of characters T −→ C and let Φ ⊂ X be the root system with ∆ = {α1 , ..., α` } a choice of simple system. Then E = X ⊗Z R is a euclidean space with an inner product h , i such that the Weyl group (W0 , S0 ) of G with respect to T acts naturally on E and preserves its inner product, where S0 is the set of simple reflections si = sαi , 1 ≤ i ≤ `. We denote by N the group of all translations Tλ (λ ∈ X) on E: Tλ sends x to x + λ,. Then the semidirect product W = W0 n N is called an affine Weyl group. Let K be the dual of the e Sometimes we denote W just by type of G. Then we define the type of W by K. e There is a canonical homomorphism from W to W0 : w 7→ w. K. ¯ Let −α0 be the highest short root in Φ. Set s0 = sα0 T−α0 with sα0 the reflection corresponding to α0 . Then S = S0 ∪ {s0 } forms a Coxeter generator set of W . Theorem 1.9. ([12, Theorem 4.8]) In the setup of 1.8, there exists a bijection

A new algorithm for finding an l.c.r. set

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c : u 7→ c(u) from the set of unipotent conjugacy classes in G to the set Cell(W ) of two-sided cells in W such that a(c(u)) = dim Bu , where u is any element in u, and dim Bu is the dimension of the variety of Borel subgroups of G containing u. §2. Alcove forms for elements in affine Weyl groups. Keep the setup of 1.8 for an affine Weyl group W . 2.1. The alcove form of an element w ∈ W is, by definition, a Φ-tuple (k(w; α))α∈Φ over Z, subject to the following conditions: (a) k(e; α) = 0 for any α ∈ Φ, where e is the identity of W ; (b) For 0 6 i 6 l, we have k(si ; α) = 0 if α 6= ±αi ; k(si ; α) = ∓1 if α = ±αi . (c) Let w0 = wsi for some 0 6 i 6 l. Then k(w0 ; α) = k(w; (α)si ) + k(si ; α), where si = si if 1 6 i 6 l, and s0 = sα0 ( see [15, Proposition 4.2] ) It is easily checked that k(w; −α) = −k(w; α) for any α ∈ Φ. Let Φ+ be the positive system of Φ containing Π. Then the Φ-tuple (k(w; α))α∈Φ is entirely determined by the Φ+ -tuple (k(w; α))α∈Φ+ . We can identify (k(w; α))α∈Φ with (k(w; α))α∈Φ+ and call the latter also the alcove form of w. Recall the definition for a left extension of an element x ∈ W in 1.3. The following results on the alcove form (k(w; α))α∈Φ of w ∈ W are known. Proposition 2.2. (see [15, Proposition 4.7]) Let w = (k(w; α))α∈Φ ∈ W . Write w = wTλ with w ∈ W0 and λ ∈ Q. (a) For α ∈ Φ+ , we have k(w; α) = 0 if (α)w −1 ∈ Φ+ and k(w; α) = −1 if (α)w −1 ∈ Φ− . (b) R(w) = {sj ∈ S | k(w; αj ) < 0}. (c) Let w0 = wsj with w ∈ W and 0 6 j 6 l. Then for any α ∈ Φ, we have

k(w0 ; α) = k(w; (α)sj ) + k(sj ; α). (d) Let w0 = (k(w0 ; α))α∈Φ ∈ W . Then w0 is a left extension of w if and only if the inequalities k(w0 ; α)k(w; α) > 0 and |k(w0 ; α)| > |k(w; α)| hold for any α ∈ Φ.

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Jian-yi Shi The following result is crucial in the proof of Theorem 3.5.

Proposition 2.3. Let x0 , x1 , ..., xr , ... be an infinite sequence of elements in W such that xi is a right extension of xi−1 for every i > 1. Then there are some q > p > 0 such that xq is a left extension of xp . Proof. By Proposition 2.2 (a)–(c), we see that there are permutations τij , i, j > 0, on the set Φ satisfying: (i) (−α)τij = −(α)τij for any α ∈ Φ; (ii) |k(xj ; α)| > |k(xi ; (α)τij )| for any α ∈ Φ and j > i; (iii) τhi τij = τhj for any h, i, j > 0. Since |Φ| < ∞, the permutation group on Φ is finite. So there exists an infinite subsequence h1 , h2 , ..., ht , ... of 1, 2, 3, ... with τ0,ha = τ0,hb for any a, b > 0. Hence |k(xha ; α)| > |k(xhb ; α)| for any a > b > 0 and α ∈ Φ. Then by the finiteness of the set Φ, there exist some q > p > 0 in {h1 , h2 , ...} such that |k(xq ; α)| > |k(xp ; α)| and k(xq ; α) · k(xp ; α) > 0 for any α ∈ Φ. This implies that xq is a left extension of xp by Proposition 2.2 (d). ¤ §3. A new algorithm for finding an l.c.r. set in a two-sided cell. 3.1. Call a non-empty set F ⊆ W distinguished if |Γ ∩ F | 6 1 for any left cell Γ of W . Call F a representative set of left cells (or an l.c.r. set for short) of W in a two-sided cell Ω if F ⊆ Ω and |Γ ∩ F | = 1 for any left cell Γ of W in Ω. 3.2. For any two-sided cell Ω of W , set F (Ω) = {z ∈ Ω | a(sz), a(zt) < a(z) for any s ∈ L(z) and t ∈ R(z)}, E(Ω) = {z ∈ Ω | a(sz) < a(z) for any s ∈ L(z)}. Clearly, F (Ω) ⊆ E(Ω). Also, w ∈ F (Ω) if and only if w−1 ∈ F (Ω). We have the following result. Lemma 3.3. (1) Any w ∈ Ω has an expression w = x · z · y for some x, y ∈ W and z ∈ F (Ω).

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(2) An element w ∈ Ω is in E(Ω) if and only if x = 1 in any expression of the form w = x · z · y with z ∈ F (Ω). Proof. If w ∈ F (Ω), then take x = y = 1 and z = w. If w ∈ / F (Ω), then by 1.3 (2), either w = s · w0 for some s ∈ L(w), or w = w0 · t for some t ∈ R(w), where w0 ∈ Ω. By applying induction on `(w), we may write w0 = x · z · y for some x, y ∈ W and some z ∈ F (Ω). Hence w is equal to either sx · z · y or x · z · yt. This implies (1). Then (2) follows by (1) and 1.3 (2), (5).

¤

The following algorithm is for finding the set E(Ω) from F (Ω). Algorithm 3.4. (3.4.1)

Set Y0 = F (Ω);

Let k > 0. Suppose that the set Yk has been found. (3.4.2)

If Yk = ∅, then the algorithm terminates;

If Yk 6= ∅, then find the set Yk+1 = {xs | x ∈ Yk ; s ∈ S \ R(x); xs ∈ E(Ω)}. S By Lemma 3.3 (2), we have E(Ω) = i>0 Yi .

(3.4.3)

Then the following result shows that Algorithm 3.4 must terminate after a finite St number of steps, that is, E(Ω) = k=0 Yk for some t ∈ N. Theorem 3.5. Let Yj , j > 0, be obtained from the set F (Ω) by Algorithm 3.4. (1) There exists some t ∈ N such that Yj 6= ∅ and Yh = ∅ for 0 6 j 6 t < h; St (2) E(Ω) = k=0 Yk . Proof. It is easily seen that if Yi = ∅ for some i > 1 then Yj = ∅ for any j > i, or equivalently, if Yi 6= ∅ for some i > 0 then Yj 6= ∅ for any 0 6 j 6 i. Since Y0 6= ∅, to prove (1), it is enough to prove the existence of an integer i > 0 with Yi = ∅. We argue by contrary. Suppose Yi 6= ∅ for any i > 0. By the finiteness for the number of left cells of W in Ω (see 1.3 (8)), there are infinite sequences w1 , w2 , · · · in E(Ω) and 0 6 i1 < i2 < · · · < in < · · · in N satisfying that (i) wj+1 is a right extension of wj for any j > 1 (see 1.3) ; (ii) w1 ∼ w2 ∼ w3 ∼ · · · ; L

L

L

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Jian-yi Shi (iii) wj ∈ Yij for j > 1. By (i) and Proposition 2.2 (b)–(c), we see that there are permutations τij ,

i, j > 1, on Φ such that |k(wj ; α)| > |k(wi ; (α)τij )| for any α ∈ Φ, j > i > 1, and that τhi τij = τhj for any h, i, j > 1 (see the proof of Proposition 2.3). Then by Propositions 2.3, there are some q > p > 1 such that wq is a proper left extension of wp . Since wp ∼ wq are in Ω, this implies that wq is not in E(Ω), L

a contradiction. This proves (1). Then (2) follows by (1) and the definition of the set E(Ω).

¤

Remark 3.6. We say that a subset K of W is left-connected, if for any x, y ∈ K, there exists a sequence of elements x0 = x, x1 , ..., xr = y in K with some r > 0 such that xi−1 x−1 ∈ S for every 1 6 i 6 r. Lusztig conjectured in [1] that any left cell i of an affine Weyl group is left-connected. Although it has been verified in many special cases (see [14], [16], [25], [26]), the conjecture is still open in general. Now the resulting set E(Ω) of Algorithm 3.4 is useful in dealing with the conjecture. In fact, to verify the left-connectedness for a left cell Γ of W in a two-sided cell Ω, we need only to construct a graph M(Γ) with Γ ∩ E(Ω) as its vertex set. We join two vertices x 6= y in Γ ∩ E(Ω) with an edge once we find a sequence of elements x0 = x, x1 , ..., xr = y in Γ with some r > 0 such that xi−1 x−1 ∈ S for any 1 6 i 6 r. i Then we complete the proof for Γ being left-connected once the graph M(Γ) we are constructing becomes connected. Example 3.7. The following example is provided by Q. Huang, one of my Ph. D. e8 be with S = {si | 0 6 i 6 8} its distinguished generator students. Let W = E set such that o(s1 s3 ) = o(s3 s4 ) = o(s2 s4 ) = o(s4 s5 ) = o(s5 s6 ) = o(s6 s7 ) = o(s7 s8 ) = o(s8 s0 ) = 3. Let Ω be the two-sided cell of W containing the element s2 s3 s4 s2 s3 s4 . Then we can get the set E(Ω) by Algorithm 3.4. We observe that the elements w1 = s2 s3 s4 s2 s3 s4 · s1 s5 s4 s6 and w2 = s1 s4 s3 s1 s4 s3 · s2 s4 s5 s4 s6 and w3 = s3 s5 s4 s3 s5 s4 · s1 s2 s3 s4 s5 s6 in E(Ω) have the same right generalized τ invariants among themselves and have different right generalized τ -invariants from

A new algorithm for finding an l.c.r. set

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any other element in E(Ω). On the other hand, the element y := s2 · w2 = s3 s1 · w1 is a common left extension of both w1 and w2 . Since L(y) = {s1 , s2 , s3 } and L(w2 ) = {s1 , s3 , s4 } and L(s1 w1 ) = {s1 , s2 , s4 } and L(w1 ) = {s2 , s3 , s4 }, the sequence w1 , s1 w1 , y, w2 is contained in some left cell (say Γ) of W by Proposition 1.7. Also, the element x := s4 s2 s1 · w3 = s2 s4 s5 · y is a common left extension of both w2 and w3 . Since L(w3 ) = {s3 , s4 , s5 } and L(s1 w3 ) = {s1 , s4 , s5 } and L(s2 s1 w3 ) = {s1 , s2 , s5 } and L(x) = {s1 , s2 , s4 } and L(s5 y) = {s1 , s2 , s3 , s5 } and L(s4 s5 y) = {s1 , s4 }, we see that y, s5 y form a left primitive pair and so the sequence w2 , y, s5 y, s4 s5 y, x, s2 s1 w3 , s1 w3 , w3 is contained in Γ by Proposition 1.7. So Γ ∩ E(Ω) = {w1 , w2 , w3 } by Proposition 1.5. The graph M(Γ) with the vertex set {w1 , w2 , w3 } has the edges {w1 , w2 } and {w2 , w3 }, hence it is connected. This implies by 3.6 that Γ is left-connected. 3.8. In the remaining part of the section, we always assume that the set F (Ω) is known explicitly for a given two-sided cell Ω of W . For any x ∈ W , denote by Γx the left cell of W containing x. Take a distinguished subset F0 (Ω) of F (Ω) such that any w ∈ F0 (Ω) is a shortest element in the left cell Γw and that for any w ∈ F (Ω), there is some w0 ∈ F0 (Ω) and some x ∈ W with w0 · x a shortest element in the left cell Γw . In particular, when all the elements of F (Ω) have the same length (hence each w ∈ F (Ω) is a shortest element in the left cell Γw , see 4.6–4.7), we can take F0 (Ω) to be any maximal distinguished subset of F (Ω). Lemma 3.9. Assume that the set F0 (Ω) has been chosen for a two-sided cell Ω of W . Then any left cell Γ in Ω contains a shortest element w which has an expression of the form w = z · y for some z ∈ F0 (Ω) and y ∈ W . Proof. Let Γ, Γ0 be two left cells of W in Ω and let x ∈ Γ. We see by 1.3 (9) that there exists a sequence x0 = x, x1 , ..., xr in Ω with some r > 0 such that xr ∈ Γ0 and that xi−1 —–xi , R(xi−1 ) # R(xi ) for every 1 6 i 6 r. We claim that for any x0 ∈ Γ, there exists a sequence x00 = x0 , x01 , ..., x0r such that x0i−1 —–x0i

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Jian-yi Shi

and x0i ∼ xi for every 1 6 i 6 r. To show the claim, it is enough to consider L

the case where r = 1 and x0 —–x with L(x0 ) # L(x). Take s ∈ R(x1 ) \ R(x) and P t ∈ L(x) \ L(x0 ). Then we have ax1 6= 0 6= bx in the expressions Cx Cs = z az Cz P and Ct Cx0 = y by Cy , where az , by ∈ A. By the positivity of the coefficients of az , by in u (see [9, Subsection 3.1]), we see that cx1 6= 0 in the expression P Ct Cx0 Cs = v cv Cv (cv ∈ A). By the multiplication associativity of H, this implies that there exists some x01 ∈ W with dx01 6= 0 6= fx1 in the expressions P P Cx0 Cs = y0 dy0 Cy0 and Ct Cx01 = v0 fv0 Cv0 , where dy0 , fv0 ∈ A. By 1.3 (1), we get a(x0 ) = a(x) = a(x1 ) > a(x01 ) > a(x0 ) and hence a(x01 ) = a(x1 ). So x01 ∼ x1 by L

1.3 (2) and the fact x1 6 x01 . The claim is proved. L

Now we are ready to show our result. Take a shortest element w0 in Γ. Then w0 ∈ E(Ω). There exist some z ∈ F (Ω) and y ∈ W with w0 = z · y by Lemma 3.3 (2). By the construction of the set F0 (Ω), there exist some z 0 ∈ F0 (Ω) and y 0 ∈ W such that z 0 · y 0 ∼ z and `(z 0 · y 0 ) 6 `(z). Let y = s1 s2 · · · sr be a reduced L

expression of y with si ∈ S and let zi = zs1 s2 · · · si for any 0 6 i 6 r. Then the sequence z0 = z, z1 , ..., zr = w0 is in Ω with zi−1 —–zi and si ∈ R(zi ) \ R(zi−1 ). By the above claim, there exists some sequence x0 = z 0 · y 0 , x1 , ..., xr in Ω such that xi−1 —–xi and xi ∼ zi for any 1 6 i 6 r. By 1.3 (7), we have `(xr ) 6 `(x0 ) + r 6 L

`(z) + r = `(z · y) = `(w0 ). Since w0 is shortest in Γ and w0 ∼ xr , this forces L

`(xr ) = `(w0 ), hence `(z 0 · y 0 ) = `(z) and xi = xi−1 · si for any 1 6 i 6 r, in particular, xr = z 0 · y 0 · y, which is a required element w in the lemma.

¤

3.10. For any left cell Γ in a two-sided cell Ω of W , let n(Γ) be the length of a shortest element in Γ. Then n(Γ) is also the smallest number of `(z · y) as z · y ranges over all such expressions that z ∈ F0 (Ω) and y ∈ W and z · y ∈ Γ by Lemma 3.9. Let n(Ω) be the length of a shortest element in Ω. By modifying Algorithm 3.4, we get the following algorithm such that the resulting set forms an l.c.r. set of W in Ω (see Theorem 3.12): Algorithm 3.11.

A new algorithm for finding an l.c.r. set

13

Let X0 = {w ∈ F0 (Ω) | `(w) = n(Ω)};

(3.11.1)

For k > 0, suppose that the set Xk has been found. (3.11.2)

If Xk = ∅, then our algorithm terminates;

(3.11.3)

If Xk 6= ∅, then find the set

0 Xk+1 = {xs | x ∈ Xk , s ∈ S \ R(x), xs ∈ E(Ω)} ∪ {w ∈ F0 (Ω) | `(w) = Sk+1 0 n(Ω) + k + 1} and then take a maximal subset Xk+1 in Xk+1 such that i=0 Xi 0 is distinguished whenever Xk+1 6= ∅.

Theorem 3.12. Let E0 (Ω) :=

S k>0

Xk .

(1) E0 (Ω) ⊆ E(Ω); (2) The set E0 (Ω) forms an l.c.r. set of W in Ω; (3) Any w ∈ E0 (Ω) satisfies `(w) = n(Γw ). Proof. Assertion (1) and the distinguishedness of E0 (Ω) follows by the construction of the set E0 (Ω). So for the assertions (2)–(3), it is enough to prove that Γ ∩ E0 (Ω) contains an element w with `(w) = n(Γ) for any left cell Γ of W in Ω. By Lemma 3.9, there exists some w0 ∈ Γ with `(w0 ) = n(Γ) (hence w0 ∈ E(Ω)) and w0 = x·y for some x ∈ F0 (Ω) and y ∈ W . We want to find some w ∈ E0 (Ω)∩Γ with `(w) = n(Γ). Applying induction on n(Γ) > n(Ω) (see 3.10). If n(Γ) = n(Ω) then there exists some w ∈ X0 ∩ Γ ⊆ E0 ∩ Γ by the construction of the set F0 (Ω) and Algorithm 3.11. Now assume n(Γ) > n(Ω). Let k = n(Γ)−n(Ω). If w0 ∈ F0 (Ω) then we can find some w ∈ Xk ∩Γ ⊆ E0 ∩Γ by Algorithm 3.11. If w0 = x·y ∈ / F0 (Ω), i.e., `(y) > 0, take any s ∈ R(y), then z := w0 s ∈ E(Ω). We claim that z is a shortest element in the left cell Γz . For, otherwise, there would exist some z 0 ∈ Γz with `(z 0 ) < `(z). By 1.3 (6), there is some w00 ∈ Γ with w00 —–z 0 by the facts that w0 —–z (by (1.1.2)) and z ∼ z 0 and R(w0 ) * R(z). Since L

0

00

0

s ∈ R(w ) \ R(z), we have s ∈ R(w ) \ R(z ) by 1.3 (4). Hence `(w00 ) 6 `(z 0 ) + 1 6 `(z) < `(w0 ) by 1.3 (7), contradicting the assumption of `(w0 ) = n(Γ). The claim is proved. Since `(z) < `(w0 ), we have n(Γz ) < n(Γ). By inductive hypothesis, there

14

Jian-yi Shi

exists some z0 ∈ E0 (Ω) ∩ Γz with `(z0 ) = n(Γz ) = `(z). By the same argument as above with z0 in the place of z 0 , there exists some w0 ∈ Γ with w0 —–z0 and s ∈ R(w0 ) \ R(z0 ) and `(w0 ) 6 `(z0 ) + 1 = `(z) + 1 = `(w0 ). By the assumption of `(w0 ) = n(Γ), we have `(w0 ) = `(z0 ) + 1 and n(Γ) = n(Γz ) + 1. Hence w0 = z0 · s ∈ Xk0 by 1.3 (7). By the construction of the set Xk in Algorithm 3.11 and the fact n(Γ) = `(w0 ), there must exist some element in the set Xk ∩ Γ (hence in E0 (Ω) ∩ Γ). So our result follows by induction.

¤

Remark 3.13. (1) By Theorem 3.5, there is some t0 6 t with E0 (Ω) =

St0 k=0

Xk ,

where t is given as in Theorem 3.5 (1). (2) In the case where all the elements in F0 (Ω) have the same length, we can take X0 = F0 (Ω). This is so for the most cases we have dealt with in applying Algorithm 3.11 so far in practice. §4. Some applications of Algorithm 3.11. In this section, we shall discuss some technical tricks in applying Algorithm 3.11. Let us first observe an example. e4 be with S = {s0 , s1 , s2 , s3 , s4 } its Coxeter generator Example 4.1. Let W = C set, where o(s0 s1 ) = o(s3 s4 ) = 4 and o(s1 s2 ) = o(s2 s3 ) = 3. In the subsequent discussion, we denote si by i, 0 6 i 6 4, for simplifying the notation. The set W(5) is a single two-sided cell of W by Theorem 1.9. Let x1 = x2 =

01014,

x3 =

1210124,

y1 =

34341,

y2 =

34340,

y3 =

3234320.

01013,

Then F (W(5) ) =

{xi , yj | 1 6 i, j 6 3}. Since x3 ∼ x2 21 and y3 ∼ y2 23 with `(x3 ) = `(x2 21) and L

L

`(y3 ) = `(y2 23), we can take F0 (W(5) ) = {x1 , x2 , y1 , y2 } by 3.8. By applying Algorithm 3.11, we get X0 = F0 (W(5) ). X1 = X10 = {x1 2, x2 2, y1 2, y2 2}. X2 = X20 = {x1 21, x1 23, x2 21, x2 23, y1 23, y1 21, y2 23, y2 21}. X30 = {x1 210, x1 213, x1 234, x2 210, x2 213, x2 234, y1 234, y1 231, y1 210, y2 234, y2 231, y2 210} and X3 = {x1 210, x1 213, x1 234, x2 213, x2 234, y1 234, y1 231, y1 210, y2 231, y2 210} since

A new algorithm for finding an l.c.r. set

15

x2 210 ∼ x2 2 and y2 234 ∼ y2 2. L

X40

L

= {x1 2101, x1 2310, x1 2134, x1 2343, x2 2103, x2 2132, x2 2134, y1 2343, y1 2134, y1 2310,

y1 2101, y2 2341, y2 2312, y2 2310} and X4 = {x1 2101, x1 2310, x1 2134, x1 2343, x2 2132, x2 2134, y1 2343, y1 2134, y1 2310, y1 2101,y2 2312, y2 2310} since x2 2103 ∼ x2 23 and y2 2341 ∼ y2 21. L

X50

L

= {x1 23104, x1 21343, x2 21032, x2 21034, x2 21324, y1 21340, y1 23101, y2 23412, y2 23410,

y2 23120} and X5 = {x1 23104, x1 21343, x2 21034, x2 21324, y1 21340, y1 23101, y2 23410, y2 23120} since x2 21032 ∼ x1 23 and y2 23412 ∼ y1 21. L

X60

L

= {x1 231043, x1 213432, x2 210324, x2 213243, y1 213401, y1 231012, y2 234120, y2 231201}

and X6 = {x1 231043, x1 213432, x2 213243, y1 213401, y1 231012, y2 231201} since x2 210324 ∼ L

x1 234 and y2 234120 ∼ y1 210. L

X70

= {x1 2310432, x2 2132434, y1 2134012, y2 2312010} and X7 = {x1 2310432, y1 2134012}

since x2 2132434 ∼ y1 2343 and y2 2312010 ∼ x1 2101. L

L

= {x1 23104321, y1 21340123} and X9 = X90 = {x1 231043210, y1 213401234}. S9 0 Since X10 = X10 = ∅, we see by Theorem 3.12 that E0 = i=0 Xi forms an l.c.r. X8 =

X80

set of W in W(5) with |X| = 56. 4.2. The most technical part in applying Algorithm 3.11 is to determine whether or not the element xs is in E(Ω) for any given x ∈ Xk and s ∈ S \ R(x). That is, to check the equation a(xs) = a(x) and the inequality a(rxs) < a(xs) for any r ∈ L(xs). 4.3. Checking the equation a(xs) = a(x) amounts to determining the value a(xs). The relation a(xs) > a(x) holds in general by 1.3 (5). It is helpful if one could find out all the graphs M(x) and M(xs) for any x ∈ Xk and any s ∈ S \ R(x). These graphs could be worked out efficiently by some computer programme. In the case when the graph M(x) is larger or even infinite, one need only to work out a local part M of M(x) around the vertex x. It depends for the actual size of M. Usually, we take M to be a connected subgraph with its vertex set M ⊆ M (x) satisfying that

16

Jian-yi Shi (∗)

condition Γ ∩ M (x) 6= ∅ implies Γ ∩ M 6= ∅ for any left cell Γ of W .

Call a subgraph M of M(x) representative if the vertex set M of M satisfies condition (∗). Checking a subgraph M being representative in M(x) is an easy matter: one need only to check if there always exists some z0 ∈ M satisfying z0 ∼ z for any L

y ∈ M and any z ∈ M (x) with {y, z} an edge of M(x). For any x ∈ W , the following method is efficient for finding the value a(x) in the case where a direct computation for a(x) is difficult (for example, when `(x) is larger). One may try to find a sequence x0 = x, x1 , ..., xr in W such that for every 1 6 i 6 r, the element xi si is in M (xi−1 ) with {xi , xi si } a right primitive pair for some si ∈ S and that the computation for the value a(xr ) is much easier than that for a(x) (for example, this is the case when wJ ∈ M (xr ) for some J ⊂ S). In this case, we have a(x) = a(xr ) by repeatedly applying Proposition 1.7. In practice, we often choose such a sequence x0 = x, x1 , ..., xr with `(xr ) much smaller than `(x0 ) since the value a(z) can be computed relatively easier when `(z) is getting smaller in general. When W is a finite Weyl group, one can easily get the value a(x) from the value a(w0 x) by Theorem 1.9 and by the knowledge of the special unipotent classes of the corresponding reductive algebraic group, where w0 is the longest element of W (see [7, Subsection 3.3]). 4.4. For any x ∈ Xk and any s ∈ S \ R(x) with a(xs) = a(x), checking the inequality a(rxs) < a(xs) for any r ∈ L(xs) amounts to checking if we always have y = 1 in any expression of the form xs = y · w · z with w ∈ F (Ω) and y, z ∈ W . The latter can proceed efficiently in terms of alcoves forms of elements once the set F (Ω) is given explicitly. 4.5. To find Xk+1 from the set

³S

k i=0

´ 0 Xi ∪ Xk+1 , we need to determine whether

0 or not two concerning elements x, y, with at least one of them in Xk+1 , are in the

same left cell of W .

A new algorithm for finding an l.c.r. set

17

By Propositions 1.5 and 1.7, this can proceed either by comparing their right generalized τ -invariants or with the aid of right primitive pairs. Suppose that we have got all the graphs M(x) (or their representative subgraphs) ³S ´ k 0 with x ranging over i=0 Xi ∪ Xk+1 . These data will help us in determining if two elements (say x, y) so obtained are in the same left cell: We have x ∼ y only if L

x, y have the same right generalized τ -invariants; while 1.3 (6) provides a complete invariant for the relation ∼. L

4.6. The most interesting case to apply our algorithm is when F (Ω) = {wJ ∈ Ω | J ⊆ S} 6= ∅. In this case, F (Ω) is distinguished and all the elements in F (Ω) have the same length, hence F0 (Ω) = F (Ω) by 3.8. The followings are some known cases (not being exhausted) for F (Ω) of such a form: (1) Ω is the lowest two-sided cell of W under the partial order 6 (see [16, LR

Theorem 1.1]). (2) Ω consists of fully-commutative elements (e.g., the case when the Coxeter e3 or Fe4 , and Ω contains a fullygraph of W contains no subgraph of type D4 , B commutative element) (see [24, Theorem 3.4 and Subsection 3.5]). (3) W is of simply-laced type and a(Ω) 6 6 (see [25, Theorem B]). en−1 (n > 1) and Ω corresponds to a partition λ = (λ1 , ..., λr , 1, ..., 1) (4) W is of type A of n with λr + 1 > λ1 > · · · > λr > 1 (see [20, Theorems 3.1]). el (l > 1) and a(Ω) = (l − 1)2 + 1. (5) W is of type C el (l > 2) and a(Ω) = l(l − 1). (6) W is of type B 4.7. We can describe the set F (Ω) for some two-sided cell Ω of W even when F (Ω) does not consist of elements of the form wJ , J ⊆ S. For example, when e 4 , the set W(7) = {z ∈ W | a(z) = 7} forms a single two-sided cell but W = D contains no element of the form wJ , J ⊂ S. Let s0 , s1 , s2 , s3 , s4 be the Coxeter generator set of W with s2 corresponding to the branching node of its Coxeter graph. Then F (W(7) ) = {si s2 sk si s2 si sj s2 si | i, j, k ∈ {0, 1, 3, 4} distinct} (see [5, Theorem 4.6]).

18

Jian-yi Shi It is desirable to find the sets F (Ω) for more two-sided cells Ω of W in order to

apply Algorithm 3.11. Some more technical tricks are needed in applying Algorithm 3.11. For example, Sr when the set W(i) = j=1 Ωj for some i ∈ N is a union of two-sided cells Ωj with some r > 1, sometimes we know the set F (W(i) ) := {x ∈ W(i) | a(tx), a(xs) < i, ∀ t ∈ L(x), s ∈ R(x)} but not the sets F (Ωj ) individually. Let us explain it by examples. e4 with S = {0, 1, 2, 3, 4} be as in Example 4.1. Examples 4.8. Let W = C (a) The set W(3) is a union of two two-sided cells (say Ω3,1 and Ω3,2 ) of W by Theorem 1.9. We have F (W(3) ) = {121, 232, 024} and F0 (Ω3,i ) = F (Ω3,i ). At moment, we don’t know what the set F0 (Ω3,i ) is for any i = 1, 2. So we have to assume X0 = {121, 232, 024} in applying Algorithm 3.11 to find an l.c.r. set for each of the Ω3,i , i = 1, 2. We get X10 = {1213, 1210, 2324, 2321, 0241, 0243} and X1 = {1213, 1210, 2324, 0241, 0243} since X20

2321

∼ 1213. L

= {12134, 12130, 12101, 23241, 23243, 02413, 02410, 02434} and X2 = {12134, 12130, 12101, 23243, 02413, 02410, 02434} since

X3 =

X30

23241

∼ 12134. L

= {121343, 121340, 121301, 024132, 024103, 024341}.

X4 = X40 = {1213432, 1213430, 1213014, 1213012, 0241324, 0241320, 0241034}. X5 = X50 = {12134320, 12130142, 12130143, 02413243, 02413201}. X6 = X60 = {121343201, 121301423, 121301432, 024132434, 024132010}. X7 = X70 = {1213432010, 1213014234} and X8 = X80 = ∅. Call a subset K of W right-connected, if, for any pair x, y ∈ K, there is a sequence x0 = x, x1 , ..., xr = y in K with some r > 0 such that x−1 i xi−1 ∈ S for every 1 6 i 6 r. By 1.3 (2), we see that for any i > 0 with W(i) 6= ∅, any non-empty rightconnected subset of W(i) is contained in a right cell of W and hence also in a two-sided cell of W .

A new algorithm for finding an l.c.r. set Assume

121

19

∈ Ω3,1 . Let

E1 = {121, 232, 1213, 1210, 2324, 12134, 12130, 12101, 23243, 121343, 121340, 121301, 1213432, 1213430, 1213014, 1213012, 12134320, 12130142, 12130143, 121343201, 121301423, 121301432, 1213432010, 1213014234}

and

E2 = {024, 0241, 0243, 02413, 02410, 02434, 024132, 024103, 024341, 0241324, 0241320, 0241034, 02413243, 02413201, 024132434, 024132010}

Then

121

∈ E1 and E :=

S2

i=1

S7

E0 (Ω3,i ) =

k=0

Xk = E1 ∪ E2 . We see that E2 is

a maximal right-connected subset of the set E. Also, E 0 := E1 ∪ {2321} is a union of two right-connected subsets with

1213

∼

2321

L

such that

1213

and

2321

belong

to different right-connected subsets of E 0 . This implies that E0 (Ω3,1 ) = E1 and S2 E0 (Ω3,2 ) = E2 by 1.3 (2) and by the fact that W(3) = i=1 Ω3,i . (b) The set W(4) is a union of two two-sided cells (say Ω4,1 and Ω4,2 ) of W by Theorem 1.9. Let x1 =

0101,

x2 =

1214,

x3 =

121012,

y1 =

3434,

y2 =

2320,

y3 =

Then F (W(4) ) = {xi , yj | 1 6 i, j 6 3}. Since x3 ∼ x1 21 and y3 ∼ y1 23 with L L S2 `(x1 21) = `(x3 ) and `(y1 23) = `(y3 ), we can take i=1 F0 (Ω4,i ) = {x1 , x2 , y1 , y2 }

232432.

by 3.8. Again, we don’t know what the set F0 (Ω4,i ) is for any i = 1, 2 at moment. We assume X0 = {x1 , x2 , y1 , y2 } in applying Algorithm 3.11. Then X1 = X10 = {x1 2, x2 3, x2 0, y1 2, y2 1, y2 4}. X2 = X20 = {x1 21, x1 23, x2 30, x2 32, x2 34, x2 01, y1 23, y1 21, y2 10, y2 12, y2 14, y2 43}. X30 = {x1 210, x1 213, x1 234, x2 301, x2 324, x2 304, y1 234, y1 213, y1 210, y2 143, y2 102, y2 104} and X3 = {x1 213, x1 234, x2 301, x2 324, x2 304, y1 213, y1 210, y2 143, y2 102, y2 104} since x1 210 ∼ x1 2 and y1 234 ∼ y1 2. L

X40

L

= {x1 2103, x1 2132, x1 2134, x1 2343, x2 3012, x2 3014, x2 3243, y1 2101, y1 2103, y1 2132

y1 2134, y2 1043, y2 1432, y2 1021} and X4 = {x1 2132, x1 2134, x1 2343, x2 3012, x2 3014, x2 3243, y1 2101, y1 2103, y1 2132,y2 1043, y2 1432, y2 1021} since x1 2103 ∼ x1 23 and y1 2134 ∼ y1 21. L

X50

L

= {x1 21034, x1 21324, x1 21343, x1 23432, x2 30124, y1 21012, y1 21013, y1 21032, y1 21034,

y2 10432} and X5 = {x1 21324, x1 21343, x1 23432, x2 30124, y1 21012, y1 21013, y1 21032, y2 10432} since x1 21034 ∼ x1 234 and y1 21034 ∼ y1 210. L

L

20

Jian-yi Shi X60 = {x1 210343, x1 213243, x1 213432, x1 234321, x2 301243, y1 210123, y1 210132, y1 210134,

y1 210321, y2 104321} and X6 = {x1 213243, x1 213432, x1 234321, x2 301243, y1 210123, y1 210132, y1 210321, y2 104321} since x1 210343 ∼ x1 2343 and y1 210134 ∼ y1 2101. L

X70

L

= {x1 2103432, x1 2134321, x1 2343210, x2 3012434, y1 2341012, y1 2310123, y1 2101234,

y2 1432010} and X7 = {x1 2343210, x2 3012434, y1 2101234, y2 1432010} since x1 2103432 ∼ L

x1 23432 and x1 2134321 ∼ y1 2132 and y1 2341012 ∼ y1 21012 and y1 2310123 ∼ x1 2312. L

X8 =

X80

L

L

= ∅.

Assume x1 ∈ Ω4,1 . Then by 1.3 (2), we have E0 (Ω4,1 ) = E11 ∪ E12 and E0 (Ω4,2 ) = E21 ∪ E22 , where E11 = {x1 , x1 2, x1 21, x1 23, x1 213, x1 234, x1 2132, x1 2134, x1 2343, x1 21324, x1 21343, x1 23432, x1 213243, x1 213432, x1 234321, x1 2343210}, E12 = {y1 , y1 2, y1 21, y1 23, y1 210, y1 213, y1 2101, y1 2103, y1 2132, y1 21012, y1 21013, y1 21032, y1 210123, y1 210132, y1 210321, y1 2101234}, E21 = {x2 , x2 3, x2 0, x2 30, x2 32, x2 34, x2 01, x2 301, x2 324, x2 304, x2 3012, x2 3014, x2 3243, x2 30124, x2 301243, x2 3012434} and E22 = {y2 , y2 1, y2 4, y2 10, y2 14, y2 12, y2 43, y2 102, y2 104, y2 143, y2 1043, y2 1432, y2 1021, y2 10432, y2 104321, y2 1043210}, by the following facts: (i) Each of Eij , i, j = 1, 2, is a maximal right-connected set in

S2 i,j=1

Eij ;

(ii) y2 12 ∼ y2 124 ∼ x2 and y2 12 ∈ E22 and x2 ∈ E21 ; R

L

(iii) {x1 213423, x1 2134232} forms a right primitive pair; (iv) x1 213423 ∈ E11 and y1 ∈ E12 and x1 2134232 ∼ 234232 ∼ 2342324 ∼ 234234 ∼ y1 ; L

R

R

L

(v) W(4) is a union of two two-sided cells of W . 4.9. Assume that W is an irreducible finite or affine Coxeter group of simplylaced type. We see by [25, Lemma 6.1] that if w ∈ W satisfies a(w) > 6 and a(tw), a(ws) < a(w) for any t ∈ J := L(w) and s ∈ I := R(w), then `(wJ ), `(wI ) > 6. This fact will help us to find the set F (W(7) ). Actually, all the elements of the form wJ with J ⊆ S and `(wJ ) = 7 should be in F (W(7) ); while all the other

A new algorithm for finding an l.c.r. set

21

elements w of F (W(7) ) should satisfy `(wJ ) = `(wI ) = 6 and a(tw), a(ws) < a(w) = 7 for any t ∈ J := L(w) and s ∈ I := R(w). The set F (W(k) ), k > 7, can be described similarly but involving more cases. e6 be with S = {si | 0 6 i 6 6} its Coxeter generator Example 4.10. Let W = E set, where o(s1 s3 ) = o(s3 s4 ) = o(s4 s2 ) = o(s2 s0 ) = o(s4 s5 ) = o(s5 s6 ) = 3. Then e6 by Theorem 1.9. Denote si simply by the set W(7) is a single two-sided cell of E i,

0 6 i 6 6. By the facts mentioned in 4.9, we get

F (W(7) ) = {w1346 , w1340 , w0246 , w0241 , w4561 , w4560 , w2346 , w2451 , w3450 , w13562 , w13560 , w13025 , w13026 , w02561 , w02563 , w243 · 543, w243 · 542, w345 · 243, 2031 · w342 , 5631·w345 , 5620·w245 , w245 ·345, w345 ·245, w245 ·342, w342 ·1302, w345 ·1365, w245 ·0265}.

The set F0 (W(7) ) is obtained from F (W(7) ) by removing the last nine elements since w243 · 543 ∼ w245 · 345 ∼ 5631 · w345 and w243 · 542 ∼ w345 · 245 ∼ 5620 · w245 and L

L

L

L

w345 · 243 ∼ w245 · 342 ∼ 2031 · w342 and w342 · 1302 ∼ w1340 · 20 and w345 · 1365 ∼ L

L

L

L

w4561 · 31 and w245 · 0265 ∼ w0246 · 56. L

Remark 4.11. In each of Examples 4.1, 4.8 and 4.10, the related set F (W(k) ) is given at the beginning. As the set of all two-sided minimal elements w of W(k) (i.e. w ∈ W(k) but sw, wt ∈ / W(k) for any s ∈ L(w) and t ∈ R(w)), F (W(k) ) can be found easily since it was described explicitly for all the sets W(k) , k ∈ N, of the e4 and for the set W(7) of E e6 (see [23], [27]). In general, without knowing group C the set W(k) in advance, the set F (W(k) ), k ∈ N, of any Weyl or affine Weyl group W can be found recurrently as follows. By Theorem 1.9 and the knowledge of unipotent conjugacy classes of reductive algebraic groups (see [2]), we can get the set E(W ) := {i ∈ N | W(i) 6= ∅}. For any k ∈ E(W ), suppose that the sets F (W(h)) ), h < k, have been found already. Then the set Wk | a(w) = k} by computing the a-values of

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elements in E>k . References 1. T. Asai et al., Open problems in algebraic groups, Proc. Twelfth International Symposium, Tohoku Univ., Japan (1983), 14. 2. R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, John Wiley & Sons, Ltd., 1985. 3. Y. Chen, Left cells in the Weyl group of type E8 , J. Algebra 231(2) (2000), 805–830. 4. Y. Chen and J. Y. Shi, Left cells of the Weyl group of type E7 , Comm. in Algebra 26(11) (1998), 3837–3852. e 4 , J. Algebra 128 (1990), 384–404. 5. J. Du, Cells in the affine Weyl group of type D e8 , Master Dissertation in 6. Q. Huang, Left cells of a-values 5, 6 in the affine Weyl group E East China Normal University, Shanghai, 2008. 7. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184. e7 , Master Dissertation in East 8. Y. R. Liu, Left cells of a-values 5, 6 in the affine Weyl group E China Normal University, Shanghai, 2007. 9. G. Lusztig, Cells in affine Weyl groups, in “ Algebraic Groups and Related Topics ”(R. Hotta, ed.), Advanced Studies in Pure Math., Kinokuniya and North Holland,(1985), 255–287. 10. G. Lusztig, Cells in affine Weyl groups, II, J. Algebra 109 (1987), 536–548. 11. G. Lusztig, Leading coefficients of character values of Hecke algebras, Proc. Symp. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 235–262. 12. G. Lusztig, Cells in affine Weyl groups, IV, J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 36 (1989), 297–328. ˜n and G ˜ 2 , J. Algebra 175 13. H. B. Rui, Left cells in affine Weyl groups of types other than A (1995), 732–756. 14. J. Y. Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, vol. 1179, Springer–Verlag, Lecture Notes in Math., 1986. 15. J. Y. Shi, Alcoves corresponding to an affine Weyl group, J. London Math. Soc. (2)35 (1987), 42–55. 16. J. Y. Shi, A two-sided cell in an affine Weyl group, II, J. London Math. Soc. (2)37 (1988), 253–264. 17. J. Y. Shi, A survey on the cell theory of affine Weyl groups, Advances in Science of China, Math. 3 (1990), 79–98. 18. J. Y. Shi, Left cells in affine Weyl groups, T¨ ohoku Math. J. 46 (1994), 105–124. e 4 ), Osaka J. Math. 31 (1994), 27–50. 19. J. Y. Shi, Left cells in the affine Weyl group W (D 20. J. Y. Shi, Some results relating two presentations of certain affine Weyl groups, J. Algebra 163(1) (1994), 235–257. 21. J. Y. Shi, The verification of a conjecture on the left cells in certain Coxeter group, Hiroshima Math. J. 24(3) (1994), 627–646. 22. J. Y. Shi, Left cells in the affine Weyl group of type Fe4 , J. Algebra 200 (1998), 173–206. e4 , J. Algebra 202 (1998), 745–776. 23. J. Y. Shi, Left cells in the affine Weyl group of type C 24. J. Y. Shi, Fully commutative elements and Kazhdan–Lusztig cells in the finite and affine Coxeter groups, Proc. of AMS 131 (2003), 3371–3378. 25. J. Y. Shi, Left-connectedness of some left cells in certain Coxeter groups of simply-laced type, J. Algebra 319(6) (2008), 2410–2433. ei (i = 6, 7, 8), 26. J. Y. Shi and X. G. Zhang, Left cells with a-value 4 in the affine Weyl groups E Comm. in Algebra 36(9) (2008), 3317–3346. e6 , preprint. 27. J. Y. Shi and X. G. Zhang, Left cells in the affine Weyl group E 28. C. Q. Tong, Left cells in Weyl group of type E6 , Comm. in Algebra 23 (1995), 5031–5047.

A new algorithm for finding an l.c.r. set

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e4 ), Comm. in Algebra 22(6) 29. X. F. Zhang, Cells decomposition in the affine Weyl group W (B (1994), 1955–1974.