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Inverting Kazhdan–Lusztig multiplicities to obtain simple characters

Example

In the principal block $O_{0}$ of ${s l}_{3}$ of the previous example, order the labels as $123, 213, 132, 231, 312, 321$ ; the inverse of the zeta matrix $A = P (1)$ is the integer matrix $A^{- 1} = (\begin{array}{cccccc} 1 & - 1 & - 1 & 1 & 1 & - 1 \\ 0 & 1 & 0 & - 1 & - 1 & 1 \\ 0 & 0 & 1 & - 1 & - 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}),$ and the character formula gives, with the sum supported on $x \geq w$ , $c h L (123 \cdot 0) = c h M (123 \cdot 0) - c h M (213 \cdot 0) - c h M (132 \cdot 0) + c h M (231 \cdot 0) + c h M (312 \cdot 0) - c h M (321 \cdot 0),$ $c h L (213 \cdot 0) = c h M (213 \cdot 0) - c h M (231 \cdot 0) - c h M (312 \cdot 0) + c h M (321 \cdot 0),$ $c h L (132 \cdot 0) = c h M (132 \cdot 0) - c h M (231 \cdot 0) - c h M (312 \cdot 0) + c h M (321 \cdot 0),$ $c h L (231 \cdot 0) = c h M (231 \cdot 0) - c h M (321 \cdot 0), c h L (312 \cdot 0) = c h M (312 \cdot 0) - c h M (321 \cdot 0), c h L (321 \cdot 0) = c h M (321 \cdot 0) .$ In particular the inverse coefficients are signed integers even though all Kazhdan–Lusztig polynomials are $1$ , and for $w = 123$ the formula is the BGG Euler-characteristic expression $c h L (0) = \sum_{w} (- 1)^{ℓ (w)} c h M (w \cdot 0)$ of the trivial module.

Facts & Assumptions

Given: The Axiom of Choice (inherited from the character and Euler-character suppliers below), the regular integral principal block $O_{0}$ of ${s l}_{3}$ , the label order $123, 213, 132, 231, 312, 321$ , the multiplicity matrix $A = (P_{y, w} (1))$ , and the displayed integer matrix $B$ of the Example.

[F1]

The character formula (The Kazhdan–Lusztig character formula in regular integral category O): in $O_{0}$ , $c h L (w \cdot 0) = \sum_{x} q_{w, x}^{'} (1) c h M (x \cdot 0)$ , supported on $x \geq w$ , where $q_{x, w}^{'}$ are the inverse Kazhdan–Lusztig coefficients.

[F2]

The ${s l}_{3}$ multiplicity matrix (The Kazhdan–Lusztig multiplicity matrix for the sl_3 principal block): all comparable-pair Kazhdan–Lusztig polynomials in $S_{3}$ equal $1$ , the matrix $A = P (1) = (P_{y, w} (1))$ is the Bruhat zeta matrix of $S_{3}$ (rows $y$ , columns $w$ : $A_{y, w} = 1$ exactly when $y \leq w$ ), the Bruhat ranks are ${123}, {132, 213}, {231, 312}, {321}$ with the eight covers $(123, 132), (123, 213), (132, 231), (132, 312), (213, 231), (213, 312), (231, 321), (312, 321)$ , and the nineteen comparable pairs are the transitive closure of these covers.

[F3]

Inverse Kazhdan–Lusztig coefficients (Inverse Kazhdan–Lusztig polynomials, The Kazhdan–Lusztig inversion formula): $Q^{'} = (q_{x, w}^{'}) = P^{- 1}$ is the two-sided inverse of the Kazhdan–Lusztig coefficient matrix $P = (p_{x, w})$ , defined by the finite chain sum, so specializing at $v = 1$ gives $Q^{'} (1) = A^{- 1}$ .

[F4]

Classical normalization (Kazhdan–Lusztig polynomials in the classical $q$ -normalization): $p_{y, w} (1) = P_{y, w} (1)$ , so the entries of $P (1)$ are the classical polynomial values, integers here.

[F5]

Grothendieck group and characters (The Grothendieck group and character of O, Simple and standard bases of K0(O)): the classes of the simples and separately of the Vermas are $Z$ -bases of the Grothendieck group of the block, and the formal character is an additive map on it.

[F6]

The BGG Euler-character identity (The Euler-character identity for a finite-dimensional simple module): for a dominant integral weight $λ$ , $[L (λ)] = \sum_{w \in W} (- 1)^{ℓ (w)} [M (w \circ λ)]$ in the Grothendieck group of the linkage block.

[F7]

The Weyl character formula (The Weyl character formula): for dominant integral $λ$ , $c h L (λ) = A (λ + ρ) A (ρ)^{- 1}$ , so at $λ = 0$ one gets $c h L (0) = A (ρ) A (ρ)^{- 1} = 1$ , the character of the trivial module.

Verification

technique · direct

1.1F2F4algebra

The two matrix products. Write the displayed matrix as $B$ . The product $A B$ has entries $(A B)_{y, w} = \sum_{x : y \leq x} B_{x, w}$ , a sum over the interval $[y, \cdot]$ ; reading rows $x$ of $B$ against the comparability lists of [F2]: for $y = 123$ the six rows of $B$ sum to $(1, 0, 0, 0, 0, 0)$ ; for $y = 213$ the rows with $x \in {213, 231, 312, 321}$ sum to $(0, 1, 0, 0, 0, 0)$ ; for $y = 132$ the rows with $x \in {132, 231, 312, 321}$ sum to $(0, 0, 1, 0, 0, 0)$ ; for $y = 231$ the rows $231, 321$ sum to $(0, 0, 0, 1, 0, 0)$ ; for $y = 312$ the rows $312, 321$ sum to $(0, 0, 0, 0, 1, 0)$ ; for $y = 321$ the last row is $(0, 0, 0, 0, 0, 1)$ . Hence $A B = I$ . Since $A$ is unitriangular by [F2], it has an integral two-sided inverse (the finite geometric series for $I + (A - I)$ ), so $A B = I$ implies $B = A^{- 1}$ and $B A = I$ . Equivalently, $(B A)_{y, w} = \sum_{x : x \leq w} B_{y, x}$ sum over the lower ideal of the column label, rather than over a prefix of the displayed order; for example the row- $123$ sums at $w = 132$ and $w = 312$ are $1 - 1 = 0$ and $1 - 1 - 1 + 1 = 0$ respectively.

2.1F1F3F4step 1.1

The inverse coefficients. By [F3] the matrix of inverse Kazhdan–Lusztig coefficients is $Q^{'} = P^{- 1}$ , and by [F4] its specialization at $v = 1$ is $A^{- 1} = B$ ; hence $q_{x, w}^{'} (1) = B_{x, w}$ for all $x, w$ , the entries of $B$ being integers, some negative. Substituting the rows of $B$ into the character formula of [F1], with the support $x \geq w$ of [F2], gives exactly the six displayed identities: row $123$ is $(1, - 1, - 1, 1, 1, - 1)$ , row $213$ is $(0, 1, 0, - 1, - 1, 1)$ and row $132$ is $(0, 0, 1, - 1, - 1, 1)$ , row $231$ is $(1, 0, - 1)$ on the labels $231, 312, 321$ , row $312$ is $(1, - 1)$ on $312, 321$ , and row $321$ is $1$ .

3.1F2F5F6F7step 2.1∎

The $w = 123$ row and the Euler characteristic. On the labels ordered by [F2], the first row of $B$ is $q_{123, x}^{'} (1) = (- 1)^{ℓ (x)}$ : the values are $+ 1$ at $x = 123$ (length $0$ ), $- 1$ at $x = 213, 132$ (length $1$ ), $+ 1$ at $x = 231, 312$ (length $2$ ) and $- 1$ at $x = 321$ (length $3$ ). Hence the row- $123$ identity of step 2.1 is $c h L (0) = \sum_{x} (- 1)^{ℓ (x)} c h M (x \cdot 0)$ , the BGG Euler-characteristic expression; by [F6] applied to the dominant integral weight $0$ this is the class of $L (0)$ in the Grothendieck group of the block, and by [F7] $c h L (0) = A (ρ) A (ρ)^{- 1} = 1$ is the character of the trivial module. This independent check confirms the signed inversion: all Kazhdan–Lusztig polynomials are $1$ , yet the inverse coefficients are signed integers.

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Sources

Anders Bjoerner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer (author-hosted complete 370-page PDF) (standard reference, not scraped)
Lin Chen, lecture notes (Spring 2024), Lecture 8: Hecke algebras and the Kazhdan-Lusztig conjecture (complete 10-page lecture) (standard reference, not scraped)