ON ROOT SUBSYSTEMS AND INVOLUTIONS INS_n

Abstract

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

On embedding certain Kazhdan–Lusztig cells of $$S_n$$Sn into cells of $$S_{n+1}$$Sn+1

“…has maximal paths ((3, 1), (4, 4)), ((2, 2), (4, 4)), ((2, 2), (3, 6)), ((1, 3), (4, 4)), ((1, 3), (2, 5), (3, 6)), ((1, 3), (2, 6), (3, 6)), ((1, 4), (2,5), (3,6)), ((1, 4), (2,6), (3,6)), ((1, 4), (4, 4)), and ((1, 6), (2,6), (3,6)).…”

“…In this case, w D = [3,4,7,2,6,8,1,9,5], λ = (3, 3, 2, 1) and w J(λ) w D = [7,4,3,8,6,2,9,1,5]. The corresponding increasing subsequences in w J(λ) w D are (1, 5), (2,5), (2,9), (3,5), (3,6,9), (3,8,9), (4,6,9), (4,8,9), (4,5), and (7,8,9), and may be read directly from the tableau corresponds to d = [2,3,4,1,6,7,5,8,9] and 3,2,…”

“…Then λ * = (2, 1, 5, 1). Y (λ) = {y 1 , y 2 }, where y 1 = (1, 4)(2, 6, 3, 7, 5) and y 2 = (1, 4) (2,7,6,3,5), 4)(2, 6, 3, 7, 5), y ′ 2 = (1, 4) (2,8,9,7,6,3,5), y ′ 3 = (1, 3, 4) (2,8,9,6,5), y ′ 4 = (1, 2, 8, 9, 5, 4), and y ′ 5 = (2, 8, 9, 4, 3). Putting x 1 = 1 and x 2 = (7, 8, 9) = s 8 s 7 , we see that y ′ 1 = y 1 x 1 , y ′ 2 = y 2 x 2 , and x 1 , x 2 ∈ X ′ .…”

“…, λ r ) be an r-part composition of n. Example. If λ = (1, 2, 2, 1), then Y (λ) = {y 1 , y 2 , y 3 }, where y 1 = [1,2,5,3,6,4], y 2 = [3,1,4,2,5,6], and y 3 = [2,1,3,4,6,5]. Also, Y s (λ) = {y 1 , y 2 }.…”

“…3),(1,4),(1,6), (2, 2), (2, 5), (2, 6), (3, 1),(3,6), (4, 4)}, c D = 6 and r D = 4.Now let e 1 = [2,3,4,1,6,7,5,8,9] and e 2 =[2,5,6,1,4,7,3,8,9]. Then,…”