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

    • Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      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

       

      S RADHA1 P VANCHINATHAN1

      1. VIT University, Chennai 600 127, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2021-2022 Indian Academy of Sciences, Bengaluru.