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

@misc{khanna2026hexpansion,
  title = {{The $h$-expansion of the Plethysm $h_n[p_r]$ and Polysymmetric Expansions in $h^\otimes$ and $e^\otimes$}},
  author = {Aditya Khanna},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36181},
  eprint = {2609.36181},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36181},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The h-expansion of the Plethysm h_n[p_r] and Polysymmetric Expansions in h^⊗ and e^⊗
AU  - Aditya Khanna
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36181
UR  - https://arxiv.org/abs/2609.36181
AB  - The algebra of polysymmetric functions (PSym) is defined as the tensor product of copies of the algebra of symmetric functions (Sym) where the ith 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^⊗_τ} and {e^⊗_τ}, and interpret the results via combinatorial objects called polywrapping block tabloids. The expansions of $H, E$ and E^+ in h^⊗ and e^⊗ 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.
ER  -