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Δ–Springer varieties and Hall–Littlewood polynomials
- Part of
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- Journal:
- Forum of Mathematics, Sigma / Volume 12 / 2024
- Published online by Cambridge University Press:
- 31 January 2024, e19
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- Article
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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 
$t=0$.
