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The four weighted cells of A1×A1 give a canonical basic set at odd residue characteristic

Statement

Let $W = ⟨ s, t ∣ s^{2} = t^{2} = 1, (s t)^{2} = 1 ⟩$ , the finite Weyl group $A_{1} \times A_{1}$ , and let $L (s) = 1, L (t) = 2$ . Form its weighted Hecke algebra with coefficient $v$ and the symmetric quadratic convention $(T_{r} - v^{L (r)}) (T_{r} + v^{- L (r)}) = 0$ . Let $O = Z_{p}$ for an odd prime $p$ , let $q = 1 + p$ , and specialize $v \mapsto q$ . Write $K = Q_{p}$ , $k = F_{p}$ , and let $D$ be the decomposition matrix of the four explicit $O$ -lattices obtained from the cell quotients.

The left, right, and two-sided cells are the four singleton sets ${1}, {s}, {t}, {s t}$ . Their four cell modules are pairwise distinct one-dimensional $K$ -modules, and each has an $O$ -stable rank-one lattice. Their reductions are the four distinct simple $H_{k}$ -modules, so $D$ is the $4 \times 4$ identity matrix up to the displayed ordering. Consequently, relative to any integer-valued statistic $α$ on the generic simple modules, these four cell modules form a canonical basic set in the sense of Canonical basic sets for split Hecke specializations.

This is a finite Weyl case. In Geck's normalization, its specialization also satisfies the exact hypothesis $θ (c_{χ}) \neq 0$ for every generic irreducible character in Theorem 4.11; the proof below verifies the decomposition matrix directly.

Facts & Assumptions

Given: The weighted Coxeter and Hecke conventions in the Statement, an odd prime $p$ , and the definitions of the KL basis, cells, cell modules, and canonical basic set.

[F1]

Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order and Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank: the Coxeter presentation determines length and Bruhat order; reduced words and subwords give the finite lower intervals needed for triangularity.

[F2]

Weight functions and ordered group-algebra coefficients and Simple-generator weights extend uniquely to a weight function: a weight function is additive on length-additive products, and simple-generator labels extend exactly when they agree along finite odd-labelled edges.

[F3]

The unequal-parameter Coxeter Hecke algebra and The arbitrary-rank weighted Hecke basis and Bruhat-triangular bar: the Hecke generators satisfy their stated quadratic and braid relations, and the standard elements form a free coefficient-module basis after specialization to a commutative coefficient ring.

[F4]

Existence and uniqueness of the ordered-group unequal-parameter Kazhdan–Lusztig basis: the elements $C_{w}$ form a bar-invariant triangular basis, with $C_{s} = T_{s} + v^{- L (s)}$ for $L (s) > 0$ .

[F5]

Unequal-parameter left, right, and two-sided cells and cell quotients and Weighted cell quotients are well-defined modules and bimodules: coefficient transitions define the cell preorders, and the cell quotients are modules with bases given by their cell elements.

[F6]

Canonical basic sets for split Hecke specializations: for the chosen lattice decomposition matrix $D$ , the minimum ${\overset{ˇ}{α}}_{M}$ and the two multiplicity/order conditions define a canonical basic set.

[F7]

The canonical Hecke trace and symmetrizing form and The canonical Hecke trace is symmetric and perfect: for a finite Coxeter group, the coefficient of the identity standard basis element defines a trace with a symmetric perfect pairing, and this trace is preserved by commutative base change.

[F8]

Schur elements of a split semisimple algebra with a symmetrizing trace: for a split semisimple algebra with symmetrizing trace $τ$ , the Schur elements $c_{E}$ , when they exist, are the nonzero scalars in $τ = \sum_{E} c_{E}^{- 1} χ_{E}$ .

Proof

Given: The presentation $W$ , weights $L (s) = 1, L (t) = 2$ , the specialization $v \mapsto q = 1 + p$ , and the facts above.

1.1F1F2constructalgebra

From $(s t)^{2} = 1$ and $s^{2} = t^{2} = 1$ , $s t = (s t)^{- 1} = t s$ . Thus every word reduces to one of $1, s, t, s t$ . The homomorphism to $(Z / 2 Z)^{2}$ sending $s \mapsto (1, 0)$ , $t \mapsto (0, 1)$ respects all relators and separates these four elements, so they are distinct. Their lengths are $0, 1, 1, 2$ , respectively. Since the only finite edge has even label $2$ , the odd-edge condition in [F2] permits the unequal labels $1, 2$ ; it gives a well-defined weight function $L$ .

2.1F1F3F4step 1.1algebra

Put $C_{1} = 1$ , $C_{s} = T_{s} + v^{- 1}$ , and $C_{t} = T_{t} + v^{- 2}$ . The rank-two relations give $T_{s}^{2} = (v - v^{- 1}) T_{s} + 1$ and $T_{t}^{2} = (v^{2} - v^{- 2}) T_{t} + 1$ , so direct expansion gives $C_{s}^{2} = (v + v^{- 1}) C_{s}$ and $C_{t}^{2} = (v^{2} + v^{- 2}) C_{t}$ . The generators commute because $m (s, t) = 2$ , and $C_{s t} : = C_{s} C_{t} = C_{t} C_{s} = T_{s t} + v^{- 2} T_{s} + v^{- 1} T_{t} + v^{- 3}$ . These elements are bar invariant and have leading standard term $T_{w}$ with lower Bruhat support and negative coefficients; uniqueness in [F4] identifies them as the KL basis.

2.2F3step 1.1algebra

For $r \in {s, t}$ , set $a_{s} = q$ , $a_{t} = q^{2}$ . The specialized quadratic relation is $(T_{r} - a_{r}) (T_{r} + a_{r}^{- 1}) = 0$ . Since $a_{r} + a_{r}^{- 1} \equiv 2 (m o d p)$ , it is a unit in $O$ , so $e_{r}^{+} = (T_{r} + a_{r}^{- 1}) / (a_{r} + a_{r}^{- 1})$ and $e_{r}^{-} = (a_{r} - T_{r}) / (a_{r} + a_{r}^{- 1})$ belong to $H_{O}$ . They sum to $1$ , their product is zero by the quadratic relation, and $(T_{r} - a_{r}) e_{r}^{+} = (T_{r} + a_{r}^{- 1}) e_{r}^{-} = 0$ , so they are complementary idempotents. Every rank-one element $b + c T_{r}$ equals $(b + c a_{r}) e_{r}^{+} + (b - c a_{r}^{- 1}) e_{r}^{-}$ , making both summands free of rank one. The commuting rank-one subalgebras tensor to $H_{O}$ with the standard rank-four basis [F3], so the four products $e_{s}^{ϵ} e_{t}^{δ}$ give $H_{O} ≅ O^{4}$ and hence $H_{K} ≅ K^{4}$ , $H_{k} ≅ k^{4}$ .

3.1F3F4step 2.1algebra

The complete left-product table is determined by $C_{s} C_{1} = C_{s}$ , $C_{t} C_{1} = C_{t}$ , $C_{s}^{2} = (v + v^{- 1}) C_{s}$ , $C_{t} C_{s} = C_{s t}$ , $C_{s} C_{t} = C_{s t}$ , $C_{t}^{2} = (v^{2} + v^{- 2}) C_{t}$ , $C_{s} C_{s t} = C_{s}^{2} C_{t} = (v + v^{- 1}) C_{s t}$ , and $C_{t} C_{s t} = C_{s} C_{t}^{2} = (v^{2} + v^{- 2}) C_{s t}$ . The right table is identical because the algebra is commutative. With an arrow $y \to x$ when $C_{x}$ occurs in $C_{r} C_{y}$ or $C_{y} C_{r}$ , the only non-loop arrows are $1 \to s$ , $1 \to t$ , $s \to s t$ , and $t \to s t$ . No transition points back, so mutual comparability has four singleton classes for each left, right, and two-sided preorder.

4.1F3F4F5step 3.1algebra

In the quotient for ${1}$ , both $C_{s}$ and $C_{t}$ vanish, so $T_{s} = C_{s} - v^{- 1}$ and $T_{t} = C_{t} - v^{- 2}$ act by $- v^{- 1}$ and $- v^{- 2}$ . In the quotient for ${s}$ , $C_{t} C_{s} = C_{s t}$ vanishes while $C_{s}^{2} = (v + v^{- 1}) C_{s}$ ; hence $(T_{s}, T_{t})$ acts by $(v, - v^{- 2})$ . In the quotient for ${t}$ , $C_{s} C_{t} = C_{s t}$ vanishes, so $T_{s}$ acts by $- v^{- 1}$ ; $C_{t}^{2} = (v^{2} + v^{- 2}) C_{t}$ gives $T_{t}$ -eigenvalue $v^{2}$ . In the quotient for ${s t}$ , $C_{s} C_{s t} = (v + v^{- 1}) C_{s t}$ and $C_{t} C_{s t} = (v^{2} + v^{- 2}) C_{s t}$ , so the action is $(v, v^{2})$ . By [F5] each is a well-defined one-dimensional cell module. Their four eigenvalue pairs are distinct over $K$ , since $q + q^{- 1} \neq 0$ and $q^{2} + q^{- 2} \neq 0$ in the characteristic-zero field $K$ .

5.1F5step 4.1step 2.2

The four cell characters from step 4.1 take values in $O$ , so $Λ = O$ with the displayed scalar actions is an $H_{O}$ -stable rank-one lattice for each. Their reductions have $T_{s}, T_{t}$ eigenvalues respectively $(- 1, - 1)$ , $(1, - 1)$ , $(- 1, 1)$ , and $(1, 1)$ in $k$ . Since $p$ is odd these are four distinct characters of $H_{k} ≅ k^{4}$ ; each is simple, and each reduced lattice has composition multiplicity one for exactly its matching character. Thus $D = I_{4}$ , up to ordering.

6.1F7F8step 1.1step 2.2step 5.1algebra

Let $K_{0} = Q (v)$ be the generic coefficient field and put $U_{r} = v^{L (r)} T_{r}$ , which has eigenvalues $Q_{r} = v^{2 L (r)}$ and $- 1$ . By [F7], the canonical trace on the normalized standard basis $U_{w}$ takes value $1$ on the identity and $0$ on nonidentity elements, and its generic pairing is symmetrizing. For one rank-one factor, the algebra over $K_{0}$ is $K_{0} \times K_{0}$ , with characters $χ_{+} (U_{r}) = Q_{r}$ , $χ_{-} (U_{r}) = - 1$ . By [F8], write $τ = c_{+}^{- 1} χ_{+} + c_{-}^{- 1} χ_{-}$ . Evaluating at $1$ and $U_{r}$ gives $c_{+}^{- 1} + c_{-}^{- 1} = 1$ and $Q_{r} c_{+}^{- 1} - c_{-}^{- 1} = 0$ ; solving yields $c_{+} = 1 + Q_{r}$ and $c_{-} = 1 + Q_{r}^{- 1}$ . For $A_{1} \times A_{1}$ , the algebra and trace are tensor products of the rank-one factors, so the four Schur elements are products of one term from ${1 + v^{2}, 1 + v^{- 2}}$ and one from ${1 + v^{4}, 1 + v^{- 4}}$ . Under the residue map $v \mapsto 1$ in $k$ , each product is $4 \neq 0$ . Thus Geck's Theorem 4.11 has its exact finite-Weyl and nonvanishing hypotheses here and gives the identity decomposition matrix; step 5.1 has already proved the same conclusion directly.

7.1F6step 5.1∎

For any integer-valued $α$ , an identity decomposition matrix has exactly one nonzero entry in each column, equal to $1$ . Thus for each simple $M$ , ${\overset{ˇ}{α}}_{M}$ is the $α$ -value of its matching cell module; all other rows in that column vanish. Both conditions in [F6] hold, so the four cell modules form a canonical basic set. The claim is restricted to this explicit finite Weyl group and this odd-characteristic specialization.

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Sources

George Lusztig, Hecke Algebras with Unequal Parameters, revised arXiv text (2014) (standard reference, not scraped)
Meinolf Geck, Modular Representations of Hecke Algebras, EPFL lecture notes (2006) (standard reference, not scraped)
Anders Björner and Francesco Brenti, Combinatorics of Coxeter Groups, GTM 231 (2005) (standard reference, not scraped)