Mathematics faculty and graduate students are invited to attend the PhDPreliminary Examination of Kyle Salois. Advisor: Dr. Maria Gillespie Committee: Dr. Alexander Hulpke, Dr. Renzo Cavalieri, Dr. Dongzhou Huang Title: Higher Specht Polynomials and Tableaux Bijections for Hessenberg Varieties Abstract: The cohomology rings of regular semisimple Hessenberg varieties is only understood completely 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, illustrating the connections between the symmetric group action on these cohomology rings and the Schur expansion of chromatic symmetric functions.
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