AI 中文总结
本文刻画了对称函数环中一个非齐次基，其求值对应 rook 幺半群不可约表示特征标，并利用 Schur–Weyl 对偶揭示其与传播分拆代数及 Schur 函数 smash 积的关联。
AI 中文摘要
我们给出了对称函数环中一个非齐次基的若干刻画，该基在求值时的取值正是 rook 幺半群（对称逆半群）不可约表示的特征标值。利用 Schur–Weyl 对偶，我们证明了该基与幂对称基之间的转换系数是传播分拆代数（对偶对称逆幺半群代数）的特征标。此外，该基的结构系数等于 Schur 函数的 smash（或 Heisenberg）积中的系数。
英文摘要
We give several characterizations of an inhomogeneous basis of the ring of symmetric functions whose evaluations are the character values of the irreducible representations of the rook monoid (symmetric inverse semigroup). Using Schur--Weyl duality, we show that the transition coefficients of this basis with the power symmetric basis are the characters of the propagating partition algebra (dual symmetric inverse monoid algebra). In addition, the structure coefficients of this basis are equal to the coefficients in the smash (or Heisenberg) product of Schur functions.
Comments22 pages
