Decomposition of Polysymmetric Functions and Stack Partitions
David Martinez
TL;DR
This work extends the theory of polysymmetric functions by introducing stack partitions and a new signed homogeneous basis $\{H^+_\tau\}$, establishing explicit expansions among the non-tensor bases $\{H_\tau\}, \{E^+_\tau\}, \{E_\tau\}, \{P_\tau\}$, and the new basis $\{H^+_\tau\}$. It provides a novel combinatorial matrix-counting interpretation for products of monomial polysymmetric functions and derives Pieri-type and transition formulas that relate all bases via generating functions and determinantal identities, with the involution $\Omega$ linking dual bases. The results generalize classical symmetric-function theory to the polysymmetric setting, enabling connections to algebraic geometry through motivic measures and offering concrete tools for base-to-base computations and potential extensions to tableaux-like structures. The paper also poses open questions about inner products, skew shapes, and richer combinatorial models, highlighting directions for future development in the structure of $\mathsf{P\Lambda}$ and its geometric interpretations.
Abstract
Polysymmetric functions, introduced by Asvin G. and Andrew O'Desky, provide a new framework linking combinatorics and algebraic geometry through motivic measures in the Grothendieck ring of varieties. Building on this foundation and the combinatorial results of Khanna and Loehr, we study the non-tensor bases of the polysymmetric algebra, introducing a new basis $\{H^+_τ\}$ and deriving explicit expansions among the families $\{H_τ\}$, $\{E_τ\}$, $\{E^+_τ\}$, $\{P_τ\}$, and $\{H^+_τ\}$, where each basis is indexed by stack partitions (types) that generalize integer partitions by recording both degrees and multiplicities. Using the Pieri-type rule established by Khanna and Loehr for related families, we extend it to the $H^+$ basis, give a combinatorial description of the product of monomial polysymmetric functions, and identify an involution for the $P$ basis.