Counting $\mathbb F_q$-points of orbital varieties in ad-nilpotent ideals of type $A_n$
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2025
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| _version_ | 1866915930013958144 |
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| author | Bardestani, Mohammad Mallahi-Karai, Keivan Ram, Samrith Salmasian, Hadi |
| author_facet | Bardestani, Mohammad Mallahi-Karai, Keivan Ram, Samrith Salmasian, Hadi |
| contents | Let $\mathfrak b_n(\mathbb F_q)$ denote the Lie algebra of upper triangular $n\times n$ matrices over the finite field $\mathbb F_q$, and let $\mathfrak u_n(\mathbb F_q)$ be the nilradical of $\mathfrak b_n$. For every $\mathfrak b_n(\mathbb F_q)$-stable ideal $\mathfrak a$ of $\mathfrak u_n(\mathbb F_q)$, and every partition $μ$ of $n$, we prove two formulas for the number of elements of $\mathfrak a$ of Jordan type $μ$: the first one is the Hall scalar product of a modified Hall-Littlewood function indexed by $μ$ and a chromatic quasisymmetric function associated to $\mathfrak a$, and the second one is in terms of an explicit collection of standard tableaux.
In the special case that $\mathfrak a$ is the nilradical $\mathfrak u_Λ(\mathbb F_q)$ of the parabolic subalgebra associated to a composition $Λ$ of $n$, our first formula reduces to a result of Karp and Thomas: up to an explicit polynomial factor in $q$, the number of elements in $\mathfrak u_Λ(\mathbb F_q)$ of Jordan type $μ$ is equal to the coefficient of the monomial $\mathsf x^Λ$ in the specialization of the dual Macdonald symmetric function $\mathrm Q_{μ'}(\mathsf x;q^{-1},t)$ at $t=0$.
We give three applications: (1) a formula for the number of points of a nilpotent Hessenberg variety, (2) a formula for the number of $X\in \mathfrak u_Λ(\mathbb F_q)$ that satisfy $X^2=0$, which in the special case $Λ=(1^n)$ is different from the Kirillov-Melnikov-Ekhad-Zeilberger formula, and (3) a formula for the number of double cosets $\mathrm U_1\backslash\mathrm{GL}_n(\mathbb F_q)/\mathrm U_2$ where $\mathrm U_1$ and $\mathrm U_2$ are unipotent subgroups corresponding to two $\mathfrak b_n(\mathbb F_q)$-stable ideals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_15440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting $\mathbb F_q$-points of orbital varieties in ad-nilpotent ideals of type $A_n$ Bardestani, Mohammad Mallahi-Karai, Keivan Ram, Samrith Salmasian, Hadi Combinatorics Representation Theory 05E05, 15B33, 05A30, 20G40 Let $\mathfrak b_n(\mathbb F_q)$ denote the Lie algebra of upper triangular $n\times n$ matrices over the finite field $\mathbb F_q$, and let $\mathfrak u_n(\mathbb F_q)$ be the nilradical of $\mathfrak b_n$. For every $\mathfrak b_n(\mathbb F_q)$-stable ideal $\mathfrak a$ of $\mathfrak u_n(\mathbb F_q)$, and every partition $μ$ of $n$, we prove two formulas for the number of elements of $\mathfrak a$ of Jordan type $μ$: the first one is the Hall scalar product of a modified Hall-Littlewood function indexed by $μ$ and a chromatic quasisymmetric function associated to $\mathfrak a$, and the second one is in terms of an explicit collection of standard tableaux. In the special case that $\mathfrak a$ is the nilradical $\mathfrak u_Λ(\mathbb F_q)$ of the parabolic subalgebra associated to a composition $Λ$ of $n$, our first formula reduces to a result of Karp and Thomas: up to an explicit polynomial factor in $q$, the number of elements in $\mathfrak u_Λ(\mathbb F_q)$ of Jordan type $μ$ is equal to the coefficient of the monomial $\mathsf x^Λ$ in the specialization of the dual Macdonald symmetric function $\mathrm Q_{μ'}(\mathsf x;q^{-1},t)$ at $t=0$. We give three applications: (1) a formula for the number of points of a nilpotent Hessenberg variety, (2) a formula for the number of $X\in \mathfrak u_Λ(\mathbb F_q)$ that satisfy $X^2=0$, which in the special case $Λ=(1^n)$ is different from the Kirillov-Melnikov-Ekhad-Zeilberger formula, and (3) a formula for the number of double cosets $\mathrm U_1\backslash\mathrm{GL}_n(\mathbb F_q)/\mathrm U_2$ where $\mathrm U_1$ and $\mathrm U_2$ are unipotent subgroups corresponding to two $\mathfrak b_n(\mathbb F_q)$-stable ideals. |
| title | Counting $\mathbb F_q$-points of orbital varieties in ad-nilpotent ideals of type $A_n$ |
| topic | Combinatorics Representation Theory 05E05, 15B33, 05A30, 20G40 |
| url | https://arxiv.org/abs/2503.15440 |