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Parabolic Kazhdan–Lusztig polynomials for S_3 modulo S_2

Example

Take W=S3 with simple reflections s1,s2, let J={s1}, so WJ={1,s1} and the minimal left coset representatives are JW={e, s2, s2s1}. In the notation of The parabolic Kazhdan–Lusztig bases and polynomials the two parabolic families are computed from the ordinary Kazhdan--Lusztig coefficients of S3 (the ordinary coefficients needed below have length gap at most two and are the monomials vℓ(x)−ℓ(y)): the antispherical (sign) coefficients are

ne,e=1,ne,s2=v,ne,s2s1=0,ns2,s2=1,ns2,s2s1=v,ns2s1,s2s1=1,

all other coefficients being zero; the spherical coefficients are

me,e=1,me,s2=v,me,s2s1=v2,ms2,s2=1,ms2,s2s1=v,ms2s1,s2s1=1,

all other coefficients being zero. At v=1 the antispherical matrix is (110011001) and the spherical matrix is (111011001) in the order (e,s2,s2s1). The two families therefore already differ for this smallest nontrivial parabolic: ne,s2s1=0 while me,s2s1=v2.

Facts & Assumptions

Given: W=S3 with simple reflections s1,s2, J={s1}, WJ={1,s1}, JW={e,s2,s2s1} (Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups, The parabolic Hecke module and the antispherical standard basis), and the coefficients my,x,ny,x of The parabolic Kazhdan–Lusztig bases and polynomials.

[F1]

Ordinary coefficients: py,w=0 unless y≤w, pw,w=1, and for y<w one has py,w∈vZ[v] with leading term vℓ(w)−ℓ(y) and py,w∈vℓ(w)−ℓ(y)Z[v−2] (Existence and uniqueness of the Kazhdan–Lusztig basis); for length gap one or two this forces py,w=vℓ(w)−ℓ(y) (Kazhdan–Lusztig polynomials in the classical q-normalization).

[F2]

The Bruhat poset of S3 has ranks {e}, {s1,s2}, {s1s2,s2s1}, {w0}. The identity is below every element, each rank-one element is below both rank-two elements, and both rank-two elements are below w0. The two elements in either intermediate rank are incomparable. These comparisons follow by reduced subwords (The Bruhat order on Sn by rank inequalities, Basic properties of the Bruhat order on Sn).

[F3]

Antispherical alternating sum ny,x=∑z∈WJ(−1)ℓ(z)vℓ(z)pzy,x with ℓ(zy)=ℓ(z)+ℓ(y) for z∈WJ, y∈JW (Parabolic Kazhdan–Lusztig polynomials are alternating sums of ordinary ones (1)).

[F4]

Spherical reduction my,x=pw0Jy,w0Jx with w0J=s1 (Parabolic Kazhdan–Lusztig polynomials are alternating sums of ordinary ones (2)).

Verification

technique · direct finite computation in the normalized Hecke algebra of $S_3$, using the two reduction formulas and the ordinary coefficients of [F1]
1.1F1F2algebra

The ordinary values needed. By [F2] the comparable pairs needed below are (e,e) (gap 0), (e,s2) (gap 1), (s2,s2), (s1,s1), (s1,s1s2) (gap 1), (s1s2,s1s2), (e,s2s1) (gap 2), (s1,s2s1) (gap 1), (s2,s2s1) (gap 1), (s2s1,s2s1), (s1,w0) (gap 2), (s1s2,w0) (gap 1), (s2,w0) (gap 2), (w0,w0); by [F1] the values are 1 on the diagonal, v on the listed gap-one pairs, v2 on the listed gap-two pairs, and zero on non-comparable pairs such as (s1,s2), (s1s2,s2s1), (s1s2,s2), (s1,w0)'s reverse orders, and all pairs y>x.

2.1F1F2F3step 1.1algebra

Antispherical table. Using [F3] and step 1.1, WJ={1,s1} contributes the two terms z=e and z=s1: ne,e=pe,e=1, ne,s2=pe,s2−v ps1,s2=v−0=v, ne,s2s1=pe,s2s1−v ps1,s2s1=v2−v2=0; ns2,s2=ps2,s2−v ps1s2,s2=1−0=1 (the second pair is not comparable since ℓ(s1s2)>ℓ(s2)), ns2,s2s1=ps2,s2s1−v ps1s2,s2s1=v−0=v (the pair s1s2,s2s1 is incomparable); ns2s1,s2s1=ps2s1,s2s1−v pw0,s2s1=1−0=1 (the pair w0,s2s1 has y>x). This is the full antispherical table of the Example, and coefficients with y≰x are zero by the support clause of the basis theorem.

2.2F1F4step 1.1algebra

Spherical table. Using [F4] with w0J=s1 and step 1.1: me,e=ps1,s1=1, me,s2=ps1,s1s2=v, me,s2s1=ps1,w0=v2; ms2,s2=ps1s2,s1s2=1, ms2,s2s1=ps1s2,w0=v; ms2s1,s2s1=pw0,w0=1. This is the full spherical table of the Example.

3.1step 2.1step 2.2algebra∎

Comparison at v=1 and conclusion. Setting v=1 in steps 2.1 and 2.2 gives the two displayed integer unitriangular matrices; they differ exactly in the entry (e,s2s1), where the antispherical value is 0 and the spherical value is 1=v2∣v=1. Hence the two normalizations already differ for the smallest nontrivial parabolic, and neither table can be substituted for the other.

Remarks

The ordinary values used have length gap at most two, so they follow from the degree, parity and support clauses of the ordinary Kazhdan--Lusztig theorem; the gap-three value pe,w0 is not needed and is not asserted here. The two tables are the finite inputs of the companion counterexample cex-spherical-and-antispherical-normalizations-give-different-polynomials and of the small-category examples of this page.

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