Jonah Blasiak - A canonical basis for Garsia-Procesi modules

dmtcs:2858 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) - https://doi.org/10.46298/dmtcs.2858
A canonical basis for Garsia-Procesi modulesArticle

Authors: Jonah Blasiak 1

  • 1 Department of Computer Science, University of Chicago

We identify a subalgebra $\widehat{\mathscr{H}}^+_n$ of the extended affine Hecke algebra $\widehat{\mathscr{H}}_n$ of type $A$. The subalgebra $\widehat{\mathscr{H}}^+_n$ is a u-analogue of the monoid algebra of $\mathcal{S}_n ⋉ℤ_≥0^n$ and inherits a canonical basis from that of $\widehat{\mathscr{H}}_n$. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod $n$, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient $\mathscr{R}_1^n$ of $\widehat{\mathscr{H}}^+_n$ that is a $u$-analogue of the ring of coinvariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element $*π ∈ \widehat{\mathscr{H}}^+_n$ corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that $\mathscr{R}_1^n$ has cellular quotients $\mathscr{R}_λ$ that are $u$-analogues of the Garsia-Procesi modules $R_λ$ with left cells labeled by (a PAT version of) the $λ$ -catabolizable tableaux.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: Garsia-Procesi modules,affine Hecke algebra,canonical basis,symmetric group,k-atoms,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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