The $h$-expansion of the Plethysm $h_n[p_r]$ and Polysymmetric Expansions in $h^\otimes$ and $e^\otimes$

Source abstract

The algebra of polysymmetric functions (PSym) is defined as the tensor product of copies of the algebra of symmetric functions (Sym) where the $i$th copy has variables scaled by $i$. One way to construct a basis of PSym is to start with a basis $\{f_λ\}$ of Sym and consider all pure tensors arising from this basis. Asvin G and Andrew O'Desky described four families of non-pure tensor bases, namely $H, E, E^+$ and $P$, which we call plethystic bases. In this paper, we study the expansions of the plethystic bases into the pure-tensor bases $\{h^\otimes_τ\}$ and $\{e^\otimes_τ\}$, and interpret the results via combinatorial objects called polywrapping block tabloids. The expansions of $H, E$ and $E^+$ in $h^\otimes$ and $e^\otimes$ are found via the $h$-expansion of the plethysm $h_n[p_r]$. We compute this $h$-expansion using a statistic on words and prove it through abacus methods.