Explicit Combinatorial Formulas for Some Irreducible Characters of the $GL_k\times \mathbb{S}_n$-module of multivariate diagonal harmonics

Nancy Wallace

Explicit Combinatorial Formulas for Some Irreducible Characters of the $GL_k\times \mathbb{S}_n$-module of multivariate diagonal harmonics

Combinatorics 2019-06-21 v1 Representation Theory

Authors: Nancy Wallace

Abstract

We give an explicit combinatorial formula for some irreducible components of $GL_k\times \mathbb{S}_n$ -modules of multivariate diagonal harmonics. To this end we introduce a new path combinatorial object $T_{n,s}$ allowing us to give the formula directly in terms of Schur functions. This paper also contains formulas written in terms of Schur functions in the $q$ and $t$ variables for special cases of $\nabla(e_n)$ , $\nabla^r(e_n)$ and $\Delta'_{e_k}(e_n)$ . We also give an interpretation in term of path to the adjoint dual Pieri rule applied on these $GL_k\times \mathbb{S}_n$ -characters.

Keywords

multivariate diagonal harmonics gl_k x s_n-modules irreducible characters schur functions combinatorial formulas path combinatorics adjoint dual pieri rule

Cite

@article{arxiv.1906.08740,
  title  = {Explicit Combinatorial Formulas for Some Irreducible Characters of the $GL_k\times \mathbb{S}_n$-module of multivariate diagonal harmonics},
  author = {Nancy Wallace},
  journal= {arXiv preprint arXiv:1906.08740},
  year   = {2019}
}

Comments

This is the extended version of "S\'eminaire Lotharingien de Combinatoire: Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana)", (2019), Vol 82B, #71

Explicit Combinatorial Formulas for Some Irreducible Characters of the $GL_k\times \mathbb{S}_n$-module of multivariate diagonal harmonics

Abstract

Keywords

Cite

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