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The A2 = S3 case: Steinberg inclusion-exclusion, degree product, and reciprocity

Example

Let $W = S_{3}$ in its type $A_{2}$ Coxeter presentation with $S = {s_{1}, s_{2}}$ (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation, Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations). The library defines $S_{3}$ on ${0, 1, 2}$ ; the order-preserving relabelling $j \mapsto j + 1$ identifies it with permutations of ${1, 2, 3}$ and preserves inversion number, so we write the adjacent generators as $s_{1} = (1 2)$ and $s_{2} = (2 3)$ . This is the smallest-rank finite Coxeter system whose diagram is not a product of copies of $A_{1}$ . The example checks the finite $A_{2}$ instances of parabolic factorization, descent inclusion-exclusion and the Steinberg identity of Finite descent parabolics, parabolic factorization, the Steinberg inclusion-exclusion identity, and rational growth, and of the degree product and reciprocity of The Poincare polynomial as a product of q-integers of the basic degrees, with longest-element reciprocity. The lengths, descents and reciprocity calculations use no choice. The basic-degree comparison assumes the Axiom of Choice (The Axiom of Choice) through the degree-table supplier cited in (3).

(1) Lengths and Poincare polynomial. The six elements are $1$ (length $0$ ), $s_{1}, s_{2}$ (length $1$ ), $s_{1} s_{2}, s_{2} s_{1}$ (length $2$ ) and $w_{0} = s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2}$ (length $3$ ); hence $P_{W} (t) = 1 + 2 t + 2 t^{2} + t^{3} = [2]_{t} [3]_{t} = t^{3} P_{W} (t^{- 1})$ , with $N = ℓ (w_{0}) = 3 = ∣ Φ_{+} ∣$ (The longest element as the opposition of the chamber, and longest elements of finite parabolics (1)(ii)).

(2) Descent classes and Steinberg identity. The right descent sets are $D_{R} (1) = \emptyset$ , $D_{R} (s_{1}) = {s_{1}}$ , $D_{R} (s_{2}) = {s_{2}}$ , $D_{R} (s_{1} s_{2}) = {s_{2}}$ , $D_{R} (s_{2} s_{1}) = {s_{1}}$ , $D_{R} (w_{0}) = {s_{1}, s_{2}}$ , so $D_{S}^{S} (t) = t^{3}$ and, with the interval convention $D_{I}^{J} = {w : I \subseteq D_{R} (w) \subseteq J}$ of Length generating series, descent-class series, spherical subsets, and the multivariate descent polynomial (2), the nine interval series are $D_{\emptyset}^{\emptyset} (t) = 1, D_{\emptyset}^{{s_{i}}} (t) = 1 + t + t^{2}, D_{\emptyset}^{S} (t) = P_{W} (t) = 1 + 2 t + 2 t^{2} + t^{3}, D_{{s_{i}}}^{{s_{i}}} (t) = t + t^{2}, D_{{s_{i}}}^{S} (t) = t + t^{2} + t^{3} (i = 1, 2), D_{S}^{S} (t) = t^{3} .$ The parabolic series are $P_{W_{\emptyset}} = 1$ and $P_{W_{{s_{i}}}} = 1 + t$ . The Steinberg identity (Finite descent parabolics, parabolic factorization, the Steinberg inclusion-exclusion identity, and rational growth (4)) reads $1 - \frac{2}{1 + t} + \frac{1}{1 + 2 t + 2 t^{2} + t^{3}} = \frac{t^{3}}{1 + 2 t + 2 t^{2} + t^{3}},$ which is a polynomial identity after clearing denominators, and equivalently $1 / P_{W} (t^{- 1}) = \sum_{K} (- 1)^{∣ K ∣} / P_{W_{K}} (t)$ .

(3) Degree product. The basic degrees of $A_{2}$ are $2$ and $3$ (A regular Coxeter eigenvector determines the basic degrees: the exponent-residue identification and the complete degree tables for all finite Coxeter types (2), under the stated AC premise), so $\prod_{i} [d_{i}]_{t} = [2]_{t} [3]_{t} = P_{W} (t)$ , $∣ W ∣ = P_{W} (1) = 6$ (The cardinality $∣ A ∣$ of a finite set) and $\sum_{i} (d_{i} - 1) = 1 + 2 = 3 = N$ .

(4) Reciprocity. $t^{3} P_{W} (t^{- 1}) = t^{3} (1 + 2 t^{- 1} + 2 t^{- 2} + t^{- 3}) = t^{3} + 2 t^{2} + 2 t + 1 = P_{W} (t)$ : the coefficient sequence $(1, 2, 2, 1)$ is palindromic, in agreement with The Poincare polynomial as a product of q-integers of the basic degrees, with longest-element reciprocity (2).

Facts & Assumptions

Given: The symmetric group $W = S_{3}$ in the type $A_{2}$ presentation with simple generators $s_{1}, s_{2}$ , its length function $ℓ$ , the Axiom of Choice for the basic-degree comparison, the descent sets $D_{L}, D_{R}$ and the series $P_{A}$ , $D_{I}^{J}$ of Length generating series, descent-class series, spherical subsets, and the multivariate descent polynomial.

[L1]

Under the order-preserving relabelling $j \mapsto j + 1$ , the library's $S_{3} = Sym ({0, 1, 2})$ is identified with permutations of ${1, 2, 3}$ and its adjacent generators are $s_{1} = (1 2)$ and $s_{2} = (2 3)$ (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation). Direct composition gives the distinct one-line forms $[1, 2, 3]$ , $[2, 1, 3]$ , $[1, 3, 2]$ , $[2, 3, 1]$ , $[3, 1, 2]$ , $[3, 2, 1]$ for $1, s_{1}, s_{2}, s_{1} s_{2}, s_{2} s_{1}, w_{0} = s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2}$ , respectively. These are all one-line arrangements of three entries. In the type- $A_{2}$ presentation, canceling $s_{i}^{2}$ makes every word alternating, and the braid relation reduces every alternating word of length at least four to a shorter word; thus the six listed words exhaust the presentation. Their inversion numbers are $0, 1, 1, 2, 2, 3$ (Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations); their lengths have these same values, since the length-two words are distinct from the identity and generators, while $w_{0}$ is distinct from all words of length at most two.

[L2]

The element $w_{0} = s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2}$ has length $3$ and is longest by [L1]; the finite longest-element result gives $ℓ (w_{0}) = ∣ Φ_{+} ∣$ and $ℓ (w_{0} w) = ℓ (w_{0}) - ℓ (w)$ for all $w \in W$ (The longest element as the opposition of the chamber, and longest elements of finite parabolics (1)(ii),(iii)). Thus $N = ℓ (w_{0}) = 3 = ∣ Φ_{+} ∣$ .

[L3]

$P_{A} (t) = \sum_{w \in A} t^{ℓ (w)}$ for $A \subseteq W$ (Length generating series, descent-class series, spherical subsets, and the multivariate descent polynomial (1)).

[L4]

$D_{I}^{J} (t) = \sum_{{w : I \subseteq D_{R} (w) \subseteq J}} t^{ℓ (w)}$ uses the inclusive interval convention (Length generating series, descent-class series, spherical subsets, and the multivariate descent polynomial (2)).

[L5]

The factorization theorem gives unique length-additive parabolic factorizations; for $I = {s_{2}}$ , $W^{{s_{2}}} = {1, s_{1}, s_{2} s_{1}}$ and $W_{{s_{2}}} = {1, s_{2}}$ (Finite descent parabolics, parabolic factorization, the Steinberg inclusion-exclusion identity, and rational growth (2)).

[L6]

For all $I \subseteq J \subseteq S$ one has $D_{I}^{J} (t) = \sum_{J ∖ I \subseteq K \subseteq J} (- 1)^{∣ J ∖ K ∣} P_{W^{S ∖ K}} (t)$ with $W^{S ∖ K} = {w : D_{R} (w) \subseteq K}$ (Finite descent parabolics, parabolic factorization, the Steinberg inclusion-exclusion identity, and rational growth (3)).

[L7]

For finite $W$ , $\sum_{K \subseteq S} (- 1)^{∣ K ∣} / P_{W_{K}} (t) = t^{N} / P_{W} (t)$ (Finite descent parabolics, parabolic factorization, the Steinberg inclusion-exclusion identity, and rational growth (4)).

[L8]

Under the Axiom of Choice (The Axiom of Choice), the independently determined basic-degree table gives $d_{1} = 2, d_{2} = 3$ for $A_{2}$ (A regular Coxeter eigenvector determines the basic degrees: the exponent-residue identification and the complete degree tables for all finite Coxeter types (2)).

[L9]

The longest-element bijection and the length complementation in [L2] give $t^{N} P_{W} (t^{- 1}) = P_{W} (t)$ for finite $W$ , by summing $t^{N - ℓ (w)}$ over $w$ ; this is the choice-free argument of The Poincare polynomial as a product of q-integers of the basic degrees, with longest-element reciprocity (2), Proof 1.2.

[L10]

$P_{A_{2}} (t) = [3]_{t}!$ and $P_{S_{3}} (t) = P_{A_{2}} (t)$ , so the classical product gives $[2]_{t} [3]_{t}$ (Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models (3)).

[L11]

The cardinality of the six-element set $S_{3}$ is $6$ (The cardinality $∣ A ∣$ of a finite set).

Verification

technique · direct

1.1L1L2L3L5algebra

Length and parabolic factorization. By [L1], the six elements exhaust $W$ and have lengths $0, 1, 1, 2, 2, 3$ ; hence $P_{W} (t) = 1 + 2 t + 2 t^{2} + t^{3} = (1 + t) (1 + t + t^{2}) = [2]_{t} [3]_{t}$ and $N = ℓ (w_{0}) = 3 = ∣ Φ_{+} ∣$ by [L2]. For $I = {s_{2}}$ , [L5] gives $W^{{s_{2}}} = {1, s_{1}, s_{2} s_{1}}$ and $W_{{s_{2}}} = {1, s_{2}}$ , so the length-additive factorization gives $P_{W} = P_{W^{{s_{2}}}} P_{W_{{s_{2}}}} = (1 + t + t^{2}) (1 + t)$ , agreeing with the direct enumeration.

1.2L1L4algebra

Descent intervals. Reading right descents from the length table gives $D_{R} (1) = \emptyset$ , $D_{R} (s_{1}) = {s_{1}}$ , $D_{R} (s_{2}) = {s_{2}}$ , $D_{R} (s_{1} s_{2}) = {s_{2}}$ , $D_{R} (s_{2} s_{1}) = {s_{1}}$ and $D_{R} (w_{0}) = S$ . Thus the exact-descent series for $\emptyset, {s_{1}}, {s_{2}}, S$ are respectively $1, t + t^{2}, t + t^{2}, t^{3}$ . Summing these four classes over each inclusive interval gives $D_{\emptyset}^{\emptyset} = 1$ , $D_{\emptyset}^{{s_{i}}} = 1 + t + t^{2}$ , $D_{\emptyset}^{S} = P_{W}$ , $D_{{s_{i}}}^{{s_{i}}} = t + t^{2}$ , $D_{{s_{i}}}^{S} = t + t^{2} + t^{3}$ and $D_{S}^{S} = t^{3}$ for $i = 1, 2$ , exactly the nine cases in the statement.

2.1L4L6step 1.1step 1.2algebra

Inclusion-exclusion. Apply [L6] to the nine pairs $I \subseteq J \subseteq S$ . The needed quotients are $P_{W^{S}} = 1$ , $P_{W^{{s_{i}}}} = 1 + t + t^{2}$ (the elements whose right descents omit $s_{i}$ ), and $P_{W^{\emptyset}} = P_{W}$ . The six symmetry classes of pairs yield, respectively, $D_{\emptyset}^{\emptyset} = 1$ , $D_{\emptyset}^{{s_{i}}} = P_{W^{{s_{j}}}} = 1 + t + t^{2}$ for $j \neq i$ , $D_{\emptyset}^{S} = P_{W}$ , $D_{{s_{i}}}^{{s_{i}}} = - P_{W^{S}} + P_{W^{{s_{j}}}} = - 1 + (1 + t + t^{2}) = t + t^{2}$ , $D_{{s_{i}}}^{S} = - P_{W^{{s_{i}}}} + P_{W} = t + t^{2} + t^{3}$ , and $D_{S}^{S} = P_{W^{S}} - P_{W^{{s_{1}}}} - P_{W^{{s_{2}}}} + P_{W} = t^{3}$ . This verifies every interval value in the statement directly from the formula.

2.2step 1.1L7L9algebra

Steinberg identity. Substituting $P_{W_{\emptyset}} = 1$ , $P_{W_{{s_{1}}}} = P_{W_{{s_{2}}}} = 1 + t$ and $P_{W} = 1 + 2 t + 2 t^{2} + t^{3}$ into the finite identity of [L7] gives $1 - 2 / (1 + t) + 1 / P_{W} = t^{3} / P_{W}$ . Clearing the denominator $(1 + t) P_{W}$ yields $(1 + t) P_{W} - 2 P_{W} + (1 + t) = t^{3} (1 + t)$ ; both sides equal $t^{3} + t^{4}$ . The reciprocity of [L9] gives $t^{3} / P_{W} = 1 / P_{W} (t^{- 1})$ , so this is also the rational-function identity displayed in the statement.

3.1step 1.1L1L8L9L10L11algebra∎

Degree product and reciprocity. By [L8], the basic degrees of $A_{2}$ are $2$ and $3$ , and by [L10] the classical product is $[2]_{t} [3]_{t} = P_{W} (t)$ ; this agrees with step 1.1. At $t = 1$ , $P_{W} (1) = 6 = ∣ W ∣$ by [L11]. Also $\sum_{i} (d_{i} - 1) = 1 + 2 = 3 = N$ by step 1.1, and $t^{3} P_{W} (t^{- 1}) = t^{3} (t^{- 3} + 2 t^{- 2} + 2 t^{- 1} + 1) = 1 + 2 t + 2 t^{2} + t^{3} = P_{W} (t)$ , so the coefficient sequence $(1, 2, 2, 1)$ is palindromic as in [L9].

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Sources

A. Björner and F. Brenti, Combinatorics of Coxeter Groups, GTM 231 (class-hosted complete PDF) (standard reference, not scraped)
M. W. Davis, The Geometry and Topology of Coxeter Groups (author manuscript of the book) (standard reference, not scraped)