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| Main Authors: | , |
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| Format: | Preprint |
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2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1912.03101 |
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| _version_ | 1866929411928883200 |
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| author | Nagar, Mukesh Kumar Sivasubramanian, Sivaramakrishnan |
| author_facet | Nagar, Mukesh Kumar Sivasubramanian, Sivaramakrishnan |
| contents | Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_T^q$ and Laplacian matrix $L_T$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees on $n$ vertices. Inequalities are known between coefficients of the immanantal polynomial of $L_T$ (and $L_T^q$) as we go up the poset $GTS_n$. Using the Frobenius characteristic, this can be thought as a result involving the schur symmetric function $s_λ$. In this paper, we use an arbitrary symmetric function to define a {\it generalized matrix function} of an $n \times n$ matrix. When the symmetric function is the monomial and the forgotten symmetric function, we generalize such inequalities among coefficients of the generalized matrix polynomial of $L_T^q$ as we go up the $GTS_n$ poset. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_03101 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Generalized Matrix polynomials of Tree Laplacians indexed by Symmetric functions and the GTS poset Nagar, Mukesh Kumar Sivasubramanian, Sivaramakrishnan Combinatorics 05C05, 06A06, 15A15 Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_T^q$ and Laplacian matrix $L_T$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees on $n$ vertices. Inequalities are known between coefficients of the immanantal polynomial of $L_T$ (and $L_T^q$) as we go up the poset $GTS_n$. Using the Frobenius characteristic, this can be thought as a result involving the schur symmetric function $s_λ$. In this paper, we use an arbitrary symmetric function to define a {\it generalized matrix function} of an $n \times n$ matrix. When the symmetric function is the monomial and the forgotten symmetric function, we generalize such inequalities among coefficients of the generalized matrix polynomial of $L_T^q$ as we go up the $GTS_n$ poset. |
| title | Generalized Matrix polynomials of Tree Laplacians indexed by Symmetric functions and the GTS poset |
| topic | Combinatorics 05C05, 06A06, 15A15 |
| url | https://arxiv.org/abs/1912.03101 |