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Generalized matrix functions, determinant and permanent
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| Mohammad Hossein Jafari1, Ali Reza Madadi2 |
1- University of Tabriz , jafari@tabrizu.ac.ir
2- University of Tabriz |
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Abstract: (535 Views) |
Introduction
Since linear and multilinear algebra has many applications in different branches of sciences, the attention of many mathematicians has been attracted to it in recent decades. The determinant and the permanent are the most important functions in linear algebra and so a generalized matrix function, which is a generalization of the determinant and the permanent, becomes significant. Generalized matrix functions connect some branches of mathematics such as theory of finite groups, representation theory of groups, graph theory and combinatorics, and linear and multilinear algebra.
Let  be the symmetric group of degree  ,  be a subgroup of , and  be a function. The function
given by
is called the generalized matrix function associated to and  . Note that if  and  , the principal character of , then  is the permanent and if and  , the alternating character of , then  is the determinant.
Results
In this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. We prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. Also we show that a generalized matrix function is the determinant if and only if it preserves the product of symmetric matrices. To be precise, we show that:
If  and is a nonzero function, then the following are equivalent:
1) or ;
2)  ;
3)  ;
where  is the permutation matrix induced by  and  .
Also if and is a nonzero function, then if and only if for all symmetric matrices  . |
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| Keywords: Generalized matrix function, Determinant, Permanent, Permutation matrix, Symmetric matrix, Irreducible character, Class function. |
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Full-Text [PDF 1435 kb]
(134 Downloads)
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Type of Study: S |
Subject:
alg
Received: 2020/07/12 | Revised: 2022/11/16 | Accepted: 2020/11/14 | Published: 2022/05/21 | ePublished: 2022/05/21
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