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mmr 2022, 8(2): 76-88 |
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Jafari M H, Madadi A R. Generalized matrix functions, determinant and permanent. mmr 2022; 8 (2) :76-88
URL: http://mmr.khu.ac.ir/article-1-3111-en.html
URL: http://mmr.khu.ac.ir/article-1-3111-en.html
1- University of Tabriz , jafari@tabrizu.ac.ir
2- University of Tabriz
2- University of Tabriz
Abstract: (1494 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 ifand is a nonzero function, then if and only if for all symmetric matrices 
.
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 .
Keywords: Generalized matrix function, Determinant, Permanent, Permutation matrix, Symmetric matrix, Irreducible character, Class function.
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
Received: 2020/07/12 | Revised: 2022/11/16 | Accepted: 2020/11/14 | Published: 2022/05/21 | ePublished: 2022/05/21
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