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The degenerate level-one Hecke–KLR map for S_3

Example

Work over $F = F_{3}$ with $q = 1$ (so $e = 3$ ), $Λ = Λ_{0}$ at level one and $d = 3$ : then $H_{3}^{Λ} ≅ F_{3} S_{3}$ (Cyclotomic Hecke algebras of level one are finite Hecke algebras, Affine and cyclotomic type-A Hecke algebras) and exactly one block is nonzero, namely $α = α_{0} + α_{1} + α_{2}$ with $e_{α} = 1$ (Residue-sequence idempotents and cyclotomic Hecke blocks). The residue sequences realised by standard tableaux all have residue multiset ${0, 1, 2}$ . The degenerate KLR generators are the elements $y_{r} = \sum_{i} (x_{r} - i_{r}) e (i)$ with $x_{r}$ the Jucys–Murphy elements of $F_{3} S_{3}$ (so $x_{1} = 0$ , $x_{2} = (12)$ , $x_{3} = (13) + (23)$ ) and $ψ_{r} = \sum_{i} φ_{r} q_{r} (i)^{- 1} e (i)$ with $φ_{r} = s_{r} + p_{r} (i)$ and $p_{r} (i) = (i_{r} - i_{r + 1} + y_{r} - y_{r + 1})^{- 1}$ on unequal components (Cyclotomic Hecke intertwiners and local power series, Hecke intertwiners produce the local KLR generators). In this block the mixed relations $ψ_{1} y_{2} e (i) = y_{1} ψ_{1} e (i) + e (i)$ for $i_{1} = i_{2}$ and $ψ_{1} y_{2} e (i) = y_{1} ψ_{1} e (i)$ for $i_{1} \neq i_{2}$ , and the quadratic relation $ψ_{1}^{2} e (i) = (y_{2} - y_{1}) e (i)$ for $i_{1} \to i_{2}$ , are explicit identities in the six-dimensional algebra $F_{3} S_{3}$ .

Verification

technique · direct

Given: The group algebra $F_{3} S_{3}$ with the transpositions

$c = (12)$ , $s = (23)$ , $a = (13)$ , the additive Jucys–Murphy elements

$x_{1} = 0$ , $x_{2} = c$ , $x_{3} = (13) + (23)$ , and the level-one conventions of the

Statement.

1.1givenalgebra

The two residue projectors and the $y$ -values. Put $C = c + a + s$ . Group multiplication in $F_{3} S_{3}$ gives $C^{2} = 0$ , because the three squares contribute $3 = 0$ and the six cross products contribute three copies of each 3-cycle; and $C c = C s = 1 + (123) + (132)$ . Put $e_{+} = (1 + c) / 2$ and $e_{-} = (1 - c) / 2$ ; since $c^{2} = 1$ and $2$ is invertible in $F_{3}$ , these are idempotents, and they are the actual joint CRT projectors $e (0, 1, 2)$ and $e (0, 2, 1)$ of the two standard residue words. With $x_{3} = C - c$ one computes $y_{1} = y_{2} = 0$ and $y_{3} = C$ : indeed $(c - 1) e_{+} = (c - 1) (1 + c) / 2 = 0$ , $(c - 2) e_{-} = (c + 1) (1 - c) / 2 = 0$ , and $x_{3} - i_{3}$ equals $C$ on both projectors. All other residue words vanish, and both projectors are nonzero by the group basis.

1.2givenalgebra

There is exactly one block. The full center has basis $1$ , $C$ , $1 + (123) + (132)$ and its augmentation ideal is spanned by the last two elements, with $(1 + (123) + (132))^{2} = 0$ and $C (1 + (123) + (132)) = 0$ ; hence the augmentation ideal is square-zero and contains no nonzero idempotent. Therefore the only central idempotents are $0$ and $1$ , so $F_{3} S_{3}$ is a single block and the two projectors combine as $e_{α} = e_{+} + e_{-} = 1$ .

2.1step 1.1step 1.2algebra∎

The $ψ$ -generators and the explicit identities. For the explicit source choice of the power series, $ψ_{1} = 0$ . On $e_{+}$ one has $p_{2} = - 1 + C$ and $q_{2} = - 1$ ; on $e_{-}$ one has $p_{2} = 1 + C$ and $q_{2} = 1$ . Thus $ψ_{2} e_{+} = (1 - s - C) e_{+}$ and $ψ_{2} e_{-} = (s + 1 + C) e_{-}$ . Direct group multiplication using $c^{2} = s^{2} = 1$ and $c s c = s c s$ gives $ψ_{2} e_{+} = e_{-} ψ_{2}$ , $ψ_{2} e_{-} = e_{+} ψ_{2}$ , $ψ_{2}^{2} e_{+} = C e_{+}$ , $ψ_{2}^{2} e_{-} = - C e_{-}$ , and $C ψ_{2} = ψ_{2} C = 0$ . These are exactly the claimed mixed and quadratic identities with $y_{3} = C$ and $y_{2} = 0$ ; the first-crossing mixed identities vanish on every surviving unequal-colour word, equal-first-colour words are absent, so the universal equal-colour formula also holds. The six group elements $1, c, s, c s, s c, c s c$ give dimension $6$ , with no primitive-block citation or Hecke-basis dimension loop hidden.

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Sources

Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, arXiv:0808.2032v3, Invent. Math. 178 (2009) 451–484 (complete 32-page article) (standard reference, not scraped)
Jun Hu and Andrew Mathas, Seminormal forms and cyclotomic quiver Hecke algebras of type A, arXiv:1304.0906v3 (published in Math. Ann.), complete 51-page article (standard reference, not scraped)
Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844v1, Bull. AMS 47 (2010) 419–481 (complete 66-page survey) (standard reference, not scraped)