let $a = (a_{i,j})_{1 leq i,j leq n}$ be an $n times n$ matrixwhere $n geq 2$. let $dt(a)$, its second immanant be the immanant corresponding to the partition $lambda_2 = 2,1^{n-2}$. let $g$ be a connected graph with blocks $b_1, b_2, ldots b_p$ and with$q$-exponential distance matrix $ed_g$. we given an explicitformula for $dt(ed_g)$ which shows that $dt(ed_g)$ is independent of the manner in ...

We show that for each permutation w containing no decreasing subsequence of length k, the Kazhdan-Lusztig immanant Immw(x) vanishes on all matrices having k equal rows or columns. Also, we define two filtrations of the vector space of immanants via products of matrix minors and pattern avoidance and use the above result to show that these filtrations are equivalent. Finally, we construct new an...

The main purpose of this article is to announce and provide supporting evidence for two conjectures about the characters of the Hecke algebra Hn(q) of type An_I' evaluated at elements of its Kazhdan-Lusztig basis. In addition, we prove a conjectured immanant inequality for Jacobi-Trudi matrices (definitions below) and show how our conjectures would imply stronger inequalities of a similar kind....

Cauchy's determinant formula (1841) involving $\det ((1-u_i v_j)^{-1})$ is a fundamental result in symmetric function theory. It has been extended several directions, including determinantal extension by Frobenius [J. reine angew. Math. 1882] sum of two geometric series $u_i v_j$. This theme also resurfaced matrix analysis setting paper Horn [Trans. Amer. Soc. 1969] - where the computations are...