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Bivariate Affine q-Krawtchouk Polynomials and Association Schemes over Galois Rings

  • Yuta Watanabe
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Abstract

We introduce regularized bivariate affine q-Krawtchouk polynomials and show that they give the first eigenmatrix of a translation association scheme on Matd× n(GR(p2,r)). The relations are defined by Smith type, and the corresponding eigenvalues are written as character sums over Smith type classes. The proof is based on recurrence relations obtained from the transition numbers describing how Smith types change when one row and one column are added to a matrix. These recurrences identify the character sums with the regularized bivariate affine q-Krawtchouk polynomials.

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Cite

BibTeXRIS
BibTeX
@misc{watanabe2026bivariate,
  title = {{Bivariate Affine $q$-Krawtchouk Polynomials and Association Schemes over Galois Rings}},
  author = {Yuta Watanabe},
  howpublished = {arXiv},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2607.21913},
  eprint = {2607.21913},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.21913},
}
RIS
TY  - UNPB
TI  - Bivariate Affine q-Krawtchouk Polynomials and Association Schemes over Galois Rings
AU  - Yuta Watanabe
PY  - 2026
DA  - 2026/07/24
PB  - arXiv
DO  - 10.48550/arxiv.2607.21913
UR  - https://arxiv.org/abs/2607.21913
AB  - We introduce regularized bivariate affine q-Krawtchouk polynomials and show that they give the first eigenmatrix of a translation association scheme on Mat_d× n(GR(p^2,r)). The relations are defined by Smith type, and the corresponding eigenvalues are written as character sums over Smith type classes. The proof is based on recurrence relations obtained from the transition numbers describing how Smith types change when one row and one column are added to a matrix. These recurrences identify the character sums with the regularized bivariate affine q-Krawtchouk polynomials.
ER  -