Kazhdan–lusztig Cells and the Murphy Basis

Geck, Meinolf

doi:10.1017/s0024611506015930

Cited by 43 publications

(75 citation statements)

References 17 publications

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“…Then, as already mentioned in Example 3.8, all the two-sided cells in W are smooth, and so we now recover a known result of Schützenberger [Sch] in this case. An elementary proof that Lusztig's P Conjectures for (W, S, ϕ) hold is given in [Ge3] (see also [GeJa,§2.8…”

Section: Bonnafé and M Geckmentioning

confidence: 99%

Conjugacy classes of involutions and Kazhdan–Lusztig cells

Bonnafé¹,

Geck

2014

Represent. Theory

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Abstract. According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group S n are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz's conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types which, combined with further recent work, settles this conjecture in general.

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Section: Bonnafé and M Geckmentioning

confidence: 99%

Conjugacy classes of involutions and Kazhdan–Lusztig cells

Bonnafé¹,

Geck

2014

Represent. Theory

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“…Then, {(7, 5, 3, 1), (8,6, 2), (9, 4)} and { (7,5,3,2), (9, 6, 1), (8,4)} are decreasing covers of type (4,3,2) for w, the first being a symmetric decreasing cover. Also, { (7,8,9), (1,2,4), (3,6), (5)} is an increasing cover of type (3, 3, 2, 1) for w.…”

Section: Coversmentioning

confidence: 99%

“…In particular, if ν has parts of the same size, it is possible to find distinct elements e and e with (ν, e) and (ν, e ) corresponding to P. , 5, 7, 3, 6, 4] and P = {(7, 6, 4), (2, 1), (5, 3)}. We may take e = [4, 6, 7, 1, 2, 3, 5] and e = [4,6,7,3,5,1,2]. Then e, e ∈ X J(ν) and P = P ν e = P ν e .…”

Section: Coversmentioning

confidence: 99%

“…The ∼ sh -equivalence classes are the two-sided Kazhdan-Lusztig cells of W , as described in [11]. (See also [6,Corollary 5.6]. )…”

Section: Coversmentioning

confidence: 99%

“…(12,2), (11,3), (7,4), (6,5), (10, 9)}, {(12, 11, 3, 2), (8, 1), (7,4), (6,5), (10, 9)}, {(12, 11, 3, 2), (7,6,5,4), (8, 1), (10, 9)}, {(12, 11, 7, 6, 5, 4, 3, 2), (8, 1), (10, 9)} and {(8, 7, 6, 5, 4, 1), (12, 11, 10, 9, 3, 2)} of z 2 of types (2, 2, 2, 2, 2, 2), (4, 2, 2, 2, 2), (4, 4, 2, 2), (8, 2, 2) and (6, 6) respectively. The element z 2 has no decreasing cover of a type which dominates the types of all its decreasing covers.…”

Section: Lemma 45 Let Z ∈ W Be An Involution and Let D ∈ X J(λ) Whementioning

confidence: 99%

See 2 more Smart Citations

ON ROOT SUBSYSTEMS AND INVOLUTIONS INS_n

Deriziotis¹,

McDonough²,

Pallikaros³

2010

Glasgow Math. J.

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Abstract.Given an involution z in W , where W is the symmetric group of degree n, we study the relation between the subsystems of a root system for W corresponding to certain decreasing subsequences of z and the two-sided Kazhdan-Lusztig cell of W containing z. We study the finite symmetric group W which is a finite Coxeter group of type A. In this case, each left cell and each right cell contains exactly one involution, and each two-sided cell contains exactly one involution of a special form-a standard parabolic involution-which we describe below. The parabolic involutions form a larger collection of involutions, each of which may be associated with the standard parabolic involution in the two-sided cell containing it in a simple combinatorial way. The Robinson-Schensted-Knuth process provides a combinatorial technique for identifying the standard parabolic involution in the same two-sided cell as a given involution. Our aim is to provide a simpler combinatorial technique for carrying out this identification for a large proportion of the involutions. Not all involutions will be covered by the technique, since they must satisfy a certain length restriction which is not satisfied by all involutions. We have computed the proportion of involutions failing this restriction for symmetric groups of degree ≤12 and found it to be <0.007. Similarly, the proportion of involutions not covered by our combinatorial technique is <0.016.Moreover, if is a system of roots for W , we show that in order to identify the twosided cell of W containing an involution z ∈ W , it suffices to consider the realizations of z as the longest element of a Young subgroup W ( ) with respect to a simple generating

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