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The Kazhdan–Lusztig bases of $S_{2}$ and $S_{3}$

Facts & Assumptions

Given: The one-based permutation groups $S_{2}, S_{3}$ , with composition as functions; left $s_{i}$ swaps values $i, i + 1$ . Write $α = v - v^{- 1}$ and $C_{w} = {\underline{H}}_{w}$ .

[F1]

The standard elements form a basis, $H_{s_{i}}^{2} = 1 - α H_{s_{i}}$ , and the normalized bar assignment sends $H_{s_{i}}$ to $H_{s_{i}} + α$ (The normalized type-A Hecke algebra and its bar involution).

[F2]

The bar assignment descends to a multiplicative semilinear involution of the quotient Hecke algebra (The Hecke bar involution is well defined).

[F3]

A bar-fixed element in $H_{w} + \sum_{y < w} v Z [v] H_{y}$ is uniquely $C_{w} = {\underline{H}}_{w}$ , and these elements form a basis (Existence and uniqueness of the Kazhdan–Lusztig basis).

[F4]

The coefficients satisfy $p_{y, w} = v^{ℓ (w) - ℓ (y)} P_{y, w} (v^{- 2})$ and $μ (y, w) = [v] p_{y, w}$ (Kazhdan–Lusztig polynomials in the classical $q$ -normalization).

[F5]

Multiplication by $C_{s}$ is the descent scalar or the ascent sum with lower $s$ -descent terms (Multiplication by a generator in the Kazhdan–Lusztig basis).

[F6]

Bruhat order has the reduced-subword characterization and is graded by inversion length (Basic properties of the Bruhat order on $S_{n}$ ).

[F7]

The $R$ -coefficient $r_{y, w}$ is the coefficient of $H_{y}$ in the standard-basis expansion of $\overline{H_{w}}$ (Bruhat intervals and the $R$ -coefficients).

Example

Work in the normalization of The normalized type-A Hecke algebra and its bar involution, write permutations in one-line notation, and put $q = v^{- 2}$ as in Kazhdan–Lusztig polynomials in the classical $q$ -normalization. (a) In $S_{2}$ : $r_{i d, 21} = v - v^{- 1}$ , ${\underline{H}}_{i d} = H_{i d}$ , ${\underline{H}}_{21} = H_{21} + v H_{i d}$ , $P_{i d, 21} (q) = 1$ , $μ (i d, 21) = 1$ and ${\underline{H}}_{21}^{2} = (v + v^{- 1}) {\underline{H}}_{21}$ . (b) In $S_{3}$ the bar images of the standard basis are $\overline{H_{132}} = H_{132} + (v - v^{- 1}) H_{123}, \overline{H_{213}} = H_{213} + (v - v^{- 1}) H_{123},$ $\overline{H_{231}} = H_{231} + (v - v^{- 1}) (H_{132} + H_{213}) + (v^{2} - 2 + v^{- 2}) H_{123},$ $\overline{H_{312}} = H_{312} + (v - v^{- 1}) (H_{132} + H_{213}) + (v^{2} - 2 + v^{- 2}) H_{123},$ $\overline{H_{321}} = H_{321} + (v - v^{- 1}) (H_{231} + H_{312}) + (v^{2} - 2 + v^{- 2}) (H_{132} + H_{213}) + (v^{3} - 2 v + 2 v^{- 1} - v^{- 3}) H_{123},$ the Kazhdan–Lusztig basis is ${\underline{H}}_{123} = H_{123}, {\underline{H}}_{132} = H_{132} + v H_{123}, {\underline{H}}_{213} = H_{213} + v H_{123},$ ${\underline{H}}_{231} = H_{231} + v (H_{132} + H_{213}) + v^{2} H_{123}, {\underline{H}}_{312} = H_{312} + v (H_{132} + H_{213}) + v^{2} H_{123},$ ${\underline{H}}_{321} = H_{321} + v (H_{231} + H_{312}) + v^{2} (H_{132} + H_{213}) + v^{3} H_{123},$ all $P_{y, w} (q)$ with $y \leq w$ equal $1$ , and $μ (y, w) = 1$ exactly when $y ⋖ w$ is a cover of the Bruhat order on $S_{3}$ (the eight covers $(123, 132)$ , $(123, 213)$ , $(132, 231)$ , $(132, 312)$ , $(213, 231)$ , $(213, 312)$ , $(231, 321)$ , $(312, 321)$ in one-line notation). (c) Multiplication checks: ${\underline{H}}_{21}^{2} = (v + v^{- 1}) {\underline{H}}_{21}$ and, in the case $s w > w$ with one $μ$ -edge, ${\underline{H}}_{132} {\underline{H}}_{231} = {\underline{H}}_{321} + {\underline{H}}_{132}$ (here $s_{2} \cdot 231 = 321$ and $s_{2} \cdot 132 = 123 < 132$ , so the sum in Multiplication by a generator in the Kazhdan–Lusztig basis has the single term $μ (132, 231) = 1$ ).

Verification

1.1F1F2F3F4F7algebra

Rank one. By [F1, F2], $\overline{H_{21}} = H_{21} + α$ and $\overline{H_{i d}} = H_{i d}$ ; by [F7], this gives $r_{i d, 21} = α = v - v^{- 1}$ . Since $α + v^{- 1} = v$ , $C : = H_{21} + v H_{i d}$ is bar-fixed. It is triangular below $H_{21}$ , so uniqueness [F3] gives $C_{21} = C$ and $C_{i d} = H_{i d}$ . From $H_{21}^{2} = 1 - α H_{21}$ in [F1], $C_{21}^{2} = (H_{21} + v)^{2} = 1 + (v + v^{- 1}) H_{21} + v^{2} = (v + v^{- 1}) (H_{21} + v) .$ Its off-diagonal coefficient is $p_{i d, 21} = v$ , so [F4] gives $P_{i d, 21} = 1$ and $μ (i d, 21) = 1$ .

1.2F1F2algebra

Every bar image in $S_{3}$ . Put $X = H_{213}$ and $Y = H_{132}$ . The reduced words give $H_{231} = X Y$ , $H_{312} = Y X$ and $H_{321} = Y X Y$ . By [F1, F2], their bar images are computed by expanding $(X + α) (Y + α)$ and its reverse for the two length-two images, and $(Y + α) (X + α) (Y + α)$ for the length-three word, replacing $Y^{2}$ by $1 - α Y$ . The latter gives $H_{321} + α (H_{231} + H_{312}) + α^{2} (X + Y) + (α^{3} + α) H_{123}$ . Since $α^{2} = v^{2} - 2 + v^{- 2}$ and $α^{3} + α = v^{3} - 2 v + 2 v^{- 1} - v^{- 3}$ , these are exactly all the displayed bar images. Multiplicativity computes the images directly; it does not assert that $\overline{H_{w}}$ itself is bar-fixed.

1.3F1F2F3F6algebra

The six KL elements. Put $A = X + v$ and $B = Y + v$ . By [F1, F2], both are bar-fixed, and $A B = H_{231} + v (H_{213} + H_{132}) + v^{2} H_{123}, B A = H_{312} + v (H_{132} + H_{213}) + v^{2} H_{123} .$ A direct expansion gives $B A B = H_{321} + v (H_{231} + H_{312}) + v^{2} H_{213} + (1 + v^{2}) H_{132} + (v + v^{3}) H_{123},$ so subtracting $B = H_{132} + v H_{123}$ gives exactly the displayed $C_{321}$ . Each of $1, A, B, A B, B A, B A B - B$ is bar-fixed and has top coefficient $1$ with lower coefficients in $v Z [v]$ ; all lower indices are below its top by [F6]. Thus uniqueness [F3] identifies all six elements.

2.1F4F6step 1.3algebra

Polynomials and covers. Reading the expansions in step 1.3 gives $p_{y, w} = v^{ℓ (w) - ℓ (y)}$ for every $y \leq w$ , so [F4] gives $P_{y, w} = 1$ and $μ (y, w) = [v] p_{y, w}$ is $1$ exactly at length difference one. By the reduced-subword criterion [F6], the Bruhat ranks in $S_{3}$ are ${123}$ , ${132, 213}$ , ${231, 312}$ , and ${321}$ , and each element in one of these layers is below every element in the next layer. Hence the covers are precisely the two edges from $123$ , the four edges from rank one to rank two, and the two edges into $321$ ; these are exactly the eight pairs listed. Grading rules out other covers.

3.1F5F6step 1.1step 2.1algebra∎

The ascent multiplication check. The subword criterion [F6] gives $[i d, 231] = {123, 132, 213, 231}$ . Left $s_{2}$ sends $231$ to $321$ , whereas it sends $132$ to $123$ and $213$ to $312$ ; thus among the strict lower indices only $132$ has a left $s_{2}$ -descent. Step 2.1 gives $μ (132, 231) = 1$ , so [F5] yields $C_{132} C_{231} = C_{321} + C_{132}$ . For comparison, left $s_{1}$ sends $231$ to $132 < 231$ , and the descent formula gives $C_{213} C_{231} = (v + v^{- 1}) C_{231}$ . The rank-one square was already proved in step 1.1.

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Sources

Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 — §3.2 (printed pp. 15–16): the normalized Hecke algebra, bar involution, triangular Kazhdan–Lusztig basis, rank-one element, and the dictionary $\underline H_x=C'_x$ and $h_{y,x}=v^{\ell(x)-\ell(y)}P_{y,x}(v^{-2})$. (standard reference, not scraped)

The Kazhdan–Lusztig bases of S2S2​ and S3S3​