AI 中文总结
该研究引入正则化二元仿射$q$-克劳特楚克多项式，证明其为$\operatorname{Mat}_{d\times n}(\operatorname{GR}(p^2,r))$上平移结合方案的首个特征矩阵，通过史密斯类型定义关系与特征值，利用递推关系完成证明。
AI 中文摘要
我们引入正则化的二元仿射$q$-克劳特楚克多项式，并表明它们给出了$\operatorname{Mat}_{d\times n}(\operatorname{GR}(p^2,r))$上平移结合方案的第一个特征矩阵。关系由史密斯类型定义，相应的特征值写成史密斯类型类上的特征和。证明基于从描述当矩阵添加一行一列时史密斯类型如何变化的转移数得到的递推关系。这些递推关系将特征和与正则化的二元仿射$q$-克劳特楚克多项式等同起来。
英文摘要
We introduce regularized bivariate affine $q$-Krawtchouk polynomials and show that they give the first eigenmatrix of a translation association scheme on $\operatorname{Mat}_{d\times n}(\operatorname{GR}(p^2,r))$. 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.
Comments50 pages, to appear in Journal of Algebraic Combinatorics
Journal refJournal of Algebraic Combinatorics 64, 31 (2026)
DOI：10.1007/s10801-026-01580-1
