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The vacuum projective is a highest-weight generator

Statement

For the standard graded quotient, $1_{Λ} = [k] \in K_{0} (R^{Λ} (0))$ is a nonzero highest-weight vector of weight $Λ$ for the normalized action. Every finite projective is a finite retract of a sum of shifted word projectives $P_{ν} = R^{Λ} (β) e (ν)$ , and $[P_{ν}] = F_{ν_{n}} \dots F_{ν_{1}} 1_{Λ}$ for right append. The stronger integral generation statement is by DIVIDED words: the classes $F_{i_{r}}^{(a_{r})} \dots F_{i_{1}}^{(a_{1})} 1_{Λ}$ span $K_{0} (R^{Λ})$ over $A$ . Thus $U_{A}^{-} 1_{Λ} = K_{0} (R^{Λ})$ . The retract statement alone is not claimed to make ordinary word classes an $A$ -spanning set.

Facts & Assumptions

Given: The standard cyclotomic quotient $R^{Λ}$ and the normalized action of Normalized cyclotomic functors give an integrable quantum action.

[F1]

The rank-zero algebra is $R^{Λ} (0) = k$ and restriction from root zero has zero target, so the vacuum class is nonzero and killed by every $E_{i}$ ; its weight is $Λ$ (The cyclotomic KLR quotient at a dominant integral weight, Cyclotomic KLR induction and restriction functors).

[F2]

The finitely many word idempotents sum to the unit of each finite root block, and every finite projective is a retract of finitely many shifted copies of their principal projectives; the append bimodule sends the word $ν$ to $ν i$ , so composition acts from the rightmost functor first (Nil-Hecke idempotents and divided-power KLR projectives).

[F3]

The quotient-cover lemma (Cyclotomic projective covers and dualities descend through the quotient) says every cyclotomic indecomposable cover is the extension of its normalized generic cover, so extension of scalars is surjective on split $K_{0}$ ; the generic negative-half categorification (The KLR category categorifies the negative quantum half) identifies generic $K_{0}$ with $U_{A}^{-}$ , generated as an $A$ -module by products of the actual divided-power projectives.

Proof

technique · check the vacuum's highest-weight property directly, realise every finite projective as a retract of word projectives, and transfer divided generation from the generic negative half through the quotient

1.1F1

The vacuum is a nonzero highest-weight vector. The rank-zero block is $R^{Λ} (0) = k$ , so $1_{Λ} = [k]$ is nonzero; restriction from root zero has zero target, so $E_{i} 1_{Λ} = 0$ for every $i$ ; and the torus acts on the root-zero component by $q^{⟨ a, Λ ⟩}$ , so its weight is $Λ$ .

2.1step 1.1F2

Word projectives and retracts. The finitely many word idempotents of a root block sum to its unit, so the regular module is the sum of the corresponding word projectives and every finite projective is a retract of a finite sum of shifted copies. The append bimodule sends the word $ν$ to $ν i$ , and since operators in a composite act from right to left, $[P_{ν}] = F_{ν_{n}} \dots F_{ν_{1}} 1_{Λ}$ . This is a statement about retracts of ordinary word projectives; it does not by itself make ordinary word classes an $A$ -spanning set.

3.1step 2.1F3∎

Divided generation. The quotient-cover lemma [F3] makes extension of scalars surjective on split $K_{0}$ , so it suffices to generate the generic $K_{0}$ by extensions of divided words. The generic negative-half categorification identifies generic $K_{0}$ with $U_{A}^{-}$ and its divided-power projectives with the divided generators; their extensions are the cyclotomic divided functors applied to the vacuum, so the classes $F_{i_{r}}^{(a_{r})} \dots F_{i_{1}}^{(a_{1})} 1_{Λ}$ span $K_{0} (R^{Λ})$ over $A$ . This does not divide by a quantum factorial: $F_{i}^{a} = [a]_{q_{i}}! F_{i}^{(a)}$ has a generally nonunit coefficient, which is exactly why the stronger generation statement must use divided words. Combining this generation with the independent normalized-action result of Normalized cyclotomic functors give an integrable quantum action gives $U_{A}^{-} 1_{Λ} = K_{0} (R^{Λ})$ without a circular appeal to the integral highest-weight theorem.

Depends on

Used by

The cyclotomic projective K₀ is the integral highest-weight module Theorem

Dependency tree · two levels

35 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

Seok-Jin Kang and Masaki Kashiwara, Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras, arXiv:1102.4677v4 (standard reference, not scraped)
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups I, Representation Theory 13 (2009) 309-347, arXiv:0803.4121v2 (standard reference, not scraped)