TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Weighted cell quotients are well-defined modules and bimodules

Statement

For every Coxeter system and weight function of The unequal-parameter Coxeter Hecke algebra, the spans $I_{\leq C}^{L}$ and $I_{< C}^{L}$ attached to a left cell are left ideals; the right-cell spans are right ideals; and the two-sided-cell spans are two-sided ideals. Therefore the quotients defined in Unequal-parameter left, right, and two-sided cells and cell quotients are left modules, right modules, or bimodules, respectively. Each quotient has an $A$ -basis consisting exactly of the images of the KL basis elements indexed by its cell. No further cell property, semisimplicity, or irreducible-module classification is asserted.

Facts & Assumptions

Given: The weighted Hecke algebra $H_{A} (W, L)$ , its KL basis, the coefficient preorders, and a cell $C$ .

[F1]

Unequal-parameter left, right, and two-sided cells and cell quotients: $x \leftarrow_{L} y$ means $C_{x}$ occurs with nonzero coefficient in $C_{s} C_{y}$ for some $s \in S$ ; $\leq_{L}$ is its finite-chain closure; the left cell quotient is the span of all $C_{x}$ with $x \leq_{L} C$ , modulo the span with $x \notin C$ . The right and two-sided definitions use right or either kind of step.

[F2]

Existence and uniqueness of the ordered-group unequal-parameter Kazhdan–Lusztig basis: ${C_{w} : w \in W}$ is an $A$ -basis; each product $C_{s} C_{y}$ and $C_{y} C_{s}$ has a finite basis expansion; and $T_{s} = C_{s} - η_{s}$ for a scalar $η_{s} \in A$ .

Proof

Given: The data in the Statement.

1.1F1F2algebra

The left spans are left ideals. If $C_{x}$ occurs in $C_{s} C_{y}$ , then $x \leftarrow_{L} y$ , hence $x \leq_{L} y$ . Therefore, if $y \leq_{L} C$ , every basis term in $C_{s} C_{y}$ also has index $\leq_{L} C$ , proving closure of $I_{\leq C}^{L}$ under left multiplication by each $C_{s}$ . If $y <_{L} C$ , each such $x$ is also outside $C$ : otherwise some $g \in C$ satisfies $g \leq_{L} x \leq_{L} y \leq_{L} g$ , forcing $y \in C$ , a contradiction. Thus $I_{< C}^{L}$ is also closed under left multiplication by each $C_{s}$ . Since $T_{s} = C_{s} - η_{s}$ , the $C_{s}$ and scalars generate $H_{A} (W, L)$ , so both spans are left ideals.

1.2F1F2algebra

The right spans are right ideals. If $C_{x}$ occurs in $C_{y} C_{s}$ , then $x \leftarrow_{R} y$ , hence $x \leq_{R} y$ . Thus right multiplication by each $C_{s}$ preserves the right numerator. If $y$ is in the right denominator and $x \in C$ , choose $g \in C$ with $y \leq_{R} g$ ; since $g$ and $x$ lie in the same cell, $g \leq_{R} x$ , while $x \leq_{R} y$ . Transitivity would make $g$ equivalent to $y$ , contradicting $y \notin C$ . So right multiplication also preserves the right denominator. The generators $T_{s} = C_{s} - η_{s}$ then show that these spans are right ideals of $H_{A} (W, L)$ .

2.1F1F2step 1.1step 1.2algebra

The two-sided spans are two-sided ideals. A two-sided preorder step is a left step or a right step. Applying the closure arguments of steps 1.1 and 1.2 to each type of step shows that left and right multiplication by every $C_{s}$ , and hence by every $T_{s}$ , preserves both $I_{\leq C}^{L R}$ and $I_{< C}^{L R}$ . They are therefore two-sided ideals.

3.1F1F2step 1.1step 1.2step 2.1∎

Quotient bases and actions. The KL basis is an $A$ -basis by [F2]. For the corresponding preorder, every $x \in C$ belongs to the numerator because $x \leq x$ ; every numerator index outside $C$ belongs to the denominator by its definition. Thus the numerator basis indices are the disjoint union of the cell $C$ and the denominator indices, so their quotient has basis exactly the images of $C_{w}$ with $w \in C$ . The induced left, right, or bimodule action is well defined because the numerator and denominator are ideals of the required sidedness, as proved in steps 1.1, 1.2, and 2.1.

Depends on

Used by

Cited to discharge well-definedness by Unequal-parameter left, right, and two-sided cells and cell quotients.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

George Lusztig, Hecke Algebras with Unequal Parameters, revised arXiv text (2014) (standard reference, not scraped)