• Dimension formula for the space of relative symmetric polynomials of $D_{n}$ with respect to any irreducible representation

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/130/0016

• # Keywords

Relative symmetric polynomials; dihedral groups; invariants; supercharacters

• # Abstract

For positive integers $d$ and $n$, the vector space $H_{d} (x_{1}, x_{2}, . . . , x_{n})$ of homogeneous polynomials of degree $d$ is a representation of the symmetric group $S_{n}$ acting by permutation of variables. Regarding this as a representation for the dihedral subgroup $D_{n}$, we prove a formula for the dimension of all the isotypical subrepresentations. Our formula is simpler than the existing one found by Zamani and Babaei (Bull. Iranian Math. Soc. 40(4) (2014) 863–874). By varying the degrees $d$ we compute the generating functions for these dimensions. Further, our formula leads us naturally to a specific supercharacter theory of $D_{n}$. It turns out to be a $\ast$-product of a specific supercharacter theory studied in depth by Fowler et al. (The Ramanujan Journal (2014)), with the unique supercharacter theory of a group of order 2.

• # Author Affiliations

1. VIT University, Chennai 600 127, India

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019

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