Séminaire Lotharingien de Combinatoire, 91B.88 (2024), 12 pp.
Kyle Salois
Higher Specht Polynomials and Tableaux Bijections for Hessenberg Varieties
Abstract.
The cohomology rings of regular semisimple Hessenberg varieties are only completely understood in some cases. One such case is when the Hessenberg function is
h = (h(1),n,...,n), and is described by Abe, Horiguchi, and Masuda in 2017. We define an alternative basis for the cohomology ring in this case, which is a higher Specht basis. We give combinatorial bijections between the monomials in this basis and sets of P-tableaux, motivated by the work of Gasharov in 2008 and Shareshian and Wachs in 2016. This bijection illustrates the connection between the symmetric group action on these cohomology rings and the Schur expansion of chromatic symmetric functions. We further use the inversion formula for P-tableaux to give a new combinatorial proof of the known Poincaré polynomial for these Hessenberg varieties.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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