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Wall-crossing compositions produce the projective with the top standard label

Statement

Assume the Axiom of Choice. For a dominant regular integral representative $λ$ and a reduced word $w = s_{1} \dots s_{k}$ , set $θ_{\underline{w}} = θ_{s_{k}} \dots θ_{s_{1}}$ , so that the standards follow right multiplication in word order. Then $θ_{\underline{w}} Δ (λ) ≅ P (w \cdot λ) \oplus ⨁_{x < w} P (x \cdot λ)^{m_{x}}, m_{x} \in Z_{\geq 0} .$ Its class is $(1 + s_{1}) \dots (1 + s_{k})$ under the standard class map, with actual subexpression multiplicities. The BGG matrix satisfies $(P (x \cdot λ) : Δ (y \cdot λ)) = [Δ (y \cdot λ) : L (x \cdot λ)]$ , vanishes unless $y \leq x$ in the Bruhat order, and has diagonal entries one. Hence the top summand $P (w \cdot λ)$ occurs once and is the only summand whose flag contains $Δ (w \cdot λ)$ ; the remaining multiplicities are determined by this triangular matrix and the product class. Consequently every indecomposable projective is a summand of a word of wall crossings applied to $Δ (λ)$ .

Facts & Assumptions

Given: The regular integral block $O_{λ}$ with dominant representative $λ$ , a reduced word $w = s_{1} \dots s_{k}$ , and the word functor $θ_{\underline{w}}$ .

[F1]

The dominant Verma $Δ (λ)$ is projective in the block: by the universal property ${Hom}_{O_{λ}} (Δ (λ), N) ≅ N_{λ}$ , and the $λ$ -weight functor is exact on the block because $λ$ is the maximal weight of its linkage class; wall crossing is exact and biadjoint, hence preserves projectives; projectives have Verma flags and decompose into finitely many indecomposables (The regular integral block and its antidominant weight, The universal property of Verma modules, Projective functors on a regular integral block, Projectives in category O have finite Verma flags, Projective covers in O are indecomposable and unique, Fitting decomposition in a finite-length abelian category).

[F2]

The class of $θ_{\underline{w}} Δ (λ)$ is $(1 + s_{1}) \dots (1 + s_{k})$ with subexpression multiplicities, and every standard factor of $θ_{\underline{w}} Δ (λ)$ has a label $\leq w$ in the Bruhat order (The wall-crossing action on the Grothendieck group is multiplication by 1+s, Wall-crossing acts on standard modules by a two-step extension).

[F3]

For a projective $P (x \cdot λ)$ the BGG reciprocal multiplicities $(P (x \cdot λ) : Δ (y \cdot λ)) = [Δ (y \cdot λ) : L (x \cdot λ)]$ vanish unless $y \leq x$ in the Bruhat order, and $(P (x \cdot λ) : Δ (x \cdot λ)) = 1$ : the nonzero-composition direction is the published label triangle, and the Bruhat form follows from strong linkage and the strong-exchange/reflection-descent criterion for the dominant orbit (BGG reciprocity, The triangular restriction on projective Verma flags, The strong linkage principle for Verma modules, Finite Weyl strong exchange and deletion, Bruhat order on a finite Weyl group, The BGG criterion for homomorphisms between Verma modules).

Proof

technique · direct

1.1F1F2

The object $Y : = θ_{\underline{w}} Δ (λ)$ is projective by [F1], and the block has finite-dimensional Hom spaces and finite decompositions, so $Y ≅ ⨁_{x} P (x \cdot λ)^{m_{x}}$ with nonnegative multiplicities; its standard-flag multiplicities are read from its class in [F2], and every standard factor has a label $\leq w$ .

2.1F3step 1.1

No summand $P (x \cdot λ)$ with $x \leq̸ w$ occurs: the diagonal entry of the BGG matrix gives $(P (x \cdot λ) : Δ (x \cdot λ)) = 1$ by [F3], so such a summand would contribute a standard factor $Δ (x \cdot λ)$ to $Y$ with $x \leq̸ w$ , contradicting step 1.1.

3.1F3step 1.1step 2.1

The full subexpression contributes $Δ (w \cdot λ)$ with multiplicity one to the product class, and no other summand $P (x \cdot λ)$ with $x \leq w$ , $x \neq w$ contributes to the multiplicity of $Δ (w \cdot λ)$ : $(P (x \cdot λ) : Δ (w \cdot λ)) \neq 0$ forces $w \leq x$ by [F3], so with $x \leq w$ it forces $x = w$ . Hence $m_{w} = 1$ , and the only summand whose flag contains $Δ (w \cdot λ)$ is $P (w \cdot λ)$ .

4.1step 3.1algebra

The remaining multiplicities $m_{x}$ , $x < w$ , are determined recursively from the product class: the coefficient of $Δ (x \cdot λ)$ in the class of $Y$ equals $m_{x} + \sum_{x < z} m_{z} (P (z \cdot λ) : Δ (x \cdot λ))$ , whose second term involves only strictly larger labels already determined; solving downward in the finite Bruhat order gives all $m_{x}$ .

5.1step 3.1∎

Every indecomposable projective is a summand of such a word: the decomposition of step 1.1 with diagonal support realizes each $P (w \cdot λ)$ as a summand of $θ_{\underline{w}} Δ (λ)$ for any reduced word $w$ , which is the stated generation statement.

Depends on

Used by

The Soergel functor is not fully faithful on all category-O objects Counterexample
Wall-crossing functors do not commute Counterexample
The class map and the wall-crossing action in rank two Example
The Soergel functor on the small regular sl₂ block Example
Wall-crossing on the projectives in the sl₂ block Example
Indecomposable projective functors are indexed by the Weyl group Theorem
Indecomposable projectives correspond to indecomposable Soergel modules Theorem
Lifting of indecomposable projectives to the deformed family Theorem
Soergel's Struktursatz: the functor V is fully faithful on projectives Theorem

Dependency tree · two levels

67 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

Arthur Garnier, The classical theory of Soergel bimodules and the link with Lusztig's conjecture, Oberwolfach Arbeitsgemeinschaft note (2022) (standard reference, not scraped)
Wolfgang Soergel, Kategorie O, perverse Garben und Moduln ueber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990) 421-445 (MPIM preprint 1989/46 full text) (standard reference, not scraped)