Δ–Springer varieties and Hall–Littlewood polynomials
Forum of Mathematics, Sigma, Tome 12 (2024)
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The $\Delta $-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $\Delta $-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb {F}_q$ and partitioning the $\Delta $-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at $t=0$.
@article{10_1017_fms_2024_1,
author = {Sean T. Griffin},
title = {\ensuremath{\Delta}{\textendash}Springer varieties and {Hall{\textendash}Littlewood} polynomials},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {12},
year = {2024},
doi = {10.1017/fms.2024.1},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.1/}
}
Sean T. Griffin. Δ–Springer varieties and Hall–Littlewood polynomials. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.1
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