Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916707027648512 |
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| author | Maglione, Joshua Voll, Christopher |
| author_facet | Maglione, Joshua Voll, Christopher |
| contents | We introduce multivariate rational generating series called Hall-Littlewood-Schubert ($\mathsf{HLS}_n$) series. They are defined in terms of polynomials related to Hall-Littlewood polynomials and semistandard Young tableaux. We show that $\mathsf{HLS}_n$ series provide solutions to a range of enumeration problems upon judicious substitutions of their variables. These include the problem to enumerate sublattices of a $p$-adic lattice according to the elementary divisor types of their intersections with the members of a complete flag of reference in the ambient lattice. This is an affine analog of the stratification of Grassmannians by Schubert varieties. Other substitutions of $\mathsf{HLS}_n$ series yield new formulae for Hecke series and $p$-adic integrals associated with symplectic $p$-adic groups, and combinatorially defined quiver representation zeta functions. $\mathsf{HLS}_n$ series are $q$-analogs of Hilbert series of Stanley-Reisner rings associated with posets arising from parabolic quotients of Coxeter groups of type $\mathsf{B}$ with the Bruhat order. Special values of coarsened $\mathsf{HLS}_n$ series yield analogs of the classical Littlewood identity for the generating functions of Schur polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration Maglione, Joshua Voll, Christopher Combinatorics Group Theory Number Theory 05A15, 05E05, 11M41, 11S80, 13F55, 14M15, 16G20, 20G25 We introduce multivariate rational generating series called Hall-Littlewood-Schubert ($\mathsf{HLS}_n$) series. They are defined in terms of polynomials related to Hall-Littlewood polynomials and semistandard Young tableaux. We show that $\mathsf{HLS}_n$ series provide solutions to a range of enumeration problems upon judicious substitutions of their variables. These include the problem to enumerate sublattices of a $p$-adic lattice according to the elementary divisor types of their intersections with the members of a complete flag of reference in the ambient lattice. This is an affine analog of the stratification of Grassmannians by Schubert varieties. Other substitutions of $\mathsf{HLS}_n$ series yield new formulae for Hecke series and $p$-adic integrals associated with symplectic $p$-adic groups, and combinatorially defined quiver representation zeta functions. $\mathsf{HLS}_n$ series are $q$-analogs of Hilbert series of Stanley-Reisner rings associated with posets arising from parabolic quotients of Coxeter groups of type $\mathsf{B}$ with the Bruhat order. Special values of coarsened $\mathsf{HLS}_n$ series yield analogs of the classical Littlewood identity for the generating functions of Schur polynomials. |
| title | Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration |
| topic | Combinatorics Group Theory Number Theory 05A15, 05E05, 11M41, 11S80, 13F55, 14M15, 16G20, 20G25 |
| url | https://arxiv.org/abs/2410.08075 |