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Thursday, April 13, 2017 - 15:30 to 16:30
Prof. S. Krishnan
IIT Bombay
Hook Immanantal and Hadamard inequalities for q-Laplacians of trees
$\textbf{Abstract:}$ Let $T$ be a tree on $n$ vertices with Laplacian matrix $L$ and $q$-Laplacian $\mathcal{ L}_q$. Let $\chi_k$ be the character of the irreducible representation of $\mathfrak{S}_n$ indexed by the hook partition $k,1^{n-k}$ and let $\overline{ d}_k(L)$ be the normalized hook immanant of $L$ corresponding to the character $\chi_k$. Inequalities for $\overline{ d}_k(L)$ as $k$ increases are known. By assigning a statistic to vertex orientations on trees, we generalize these inequalities to the $q$-analogue $\mathcal{ L}_q$ of $L$ for all $q \in \mathbb{R}$ and to the bivariate $q,t$-Laplacian $\mathcal{ L}_{q,t}$ for some values $q,t$. Our statistic based approach also generalizes several other inequalities including the changing index $k(L)$ of the Hadamard inequality for $L$, to the matrix $\mathcal{ L}_q$ and $\mathcal{ L}_{q,t}$. Thus, we extend several results about $L$ to $\mathcal{ L}_q$ which includes the case when $\mathcal{ L}_q$ is not positive semidefinite.

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