Nicholas Teff

Representations on Hessenberg Varieties and Young's Rule
dmtcs:2963 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

https://doi.org/10.46298/dmtcs.2963
Representations on Hessenberg Varieties and Young's Rule
Authors: Nicholas Teff ^{1}
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Nicholas Teff
1 Department of Mathematics [IOWA]
We combinatorially construct the complex cohomology (equivariant and ordinary) of a family of algebraic varieties called regular semisimple Hessenberg varieties. This construction is purely in terms of the Bruhat order on the symmetric group. From this a representation of the symmetric group on the cohomology is defined. This representation generalizes work of Procesi, Stembridge and Tymoczko. Here a partial answer to an open question of Tymoczko is provided in our two main result. The first states, when the variety has multiple connected components, this representation is made up by inducing through a parabolic subgroup of the symmetric group. Using this, our second result obtains, for a special family of varieties, an explicit formula for this representation via Young's rule, giving the multiplicity of the irreducible representations in terms of the classical Kostka numbers.
Abe, Hiraku; Horiguchi, Tatsuya, 2020, A Survey Of Recent Developments On Hessenberg Varieties, Springer Proceedings In Mathematics & Statistics, pp. 251279, 10.1007/9789811574511_10.
Shareshian, John; Wachs, Michelle L., 2012, Chromatic Quasisymmetric Functions And Hessenberg Varieties, Configuration Spaces, pp. 433460, 10.1007/9788876424311_20.