海森堡特征标、单三角群和斐波那契数
Heisenberg characters, unitriangular groups, and Fibonacci numbers
摘要：
设$\UT_n(\FF_q)$表示有限域上具有$q$元素的幂单$n\times n$上三角矩阵群。我们表明，$\UT_{n+1}(\FF_q)$的海森堡字符索引的晶格路径从原点到线$x+y=n$使用的步骤$(1,0), (1,1), (0,1), (1,1)$，这是以一定的方式标记的非零元素的$\FF_q$。特别地，我们证明了对于$n\geq 1$，$\UT_{n+1}(\FF_q)$的Heisenberg特征标数是$q-1$中的多项式，它的系数为$n$的第n个Fibonacci数，系数为非负整数，次数为$n$.同样，我们发现$\UT_n(\FF_q)$的海森堡超特征标数是$q-1$中的多项式，其系数是Delannoy数，其值给出Pell数的$q$ -模拟.通过对一组线性特征标作用的不动点的计数，证明了由超对角元素和为零的矩阵构成的$\UT_n(\FF_q)$子群的超特征标、不可约超特征标、Heisenberg超特征标和Heisenberg特征标的个数同样都是$q-1$中的非负整系数多项式.
英文摘要：
Let $\UT_n(\FF_q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of $\UT_{n+1}(\FF_q)$ are indexed by lattice paths from the origin to the line $x+y=n$ using the steps $(1,0), (1,1), (0,1), (1,1)$, which are labeled in a certain way by nonzero elements of $\FF_q$. In particular, we prove for $n\geq 1$ that the number of Heisenberg characters of $\UT_{n+1}(\FF_q)$ is a polynomial in $q-1$ with nonnegative integer coefficients and degree $n$, whose leading coefficient is the $n$th Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of $\UT_n(\FF_q)$ is a polynomial in $q-1$ whose coefficients are Delannoy numbers and whose values give a $q$-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of $\UT_n(\FF_q)$ consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in $q-1$ with nonnegative integer coefficients.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
