Quasi-immanants

IF 1 3区 数学 Q1 MATHEMATICS
John M. Campbell
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引用次数: 0

Abstract

For an integer partition λ of n and an n×n matrix A, consider the expansion of the immanant Immλ(A) as a sum indexed by permutations σ of order n, with coefficients given by the irreducible characters χλ(ctype(σ)) of the symmetric group Sn, for the cycle type ctype(σ)n of σ. Skandera et al. have introduced combinatorial interpretations of a generalization of immanants given by replacing the coefficient χλ(ctype(σ)) with preimages with respect to the Frobenius morphism of elements among the distinguished bases of the algebra Sym of symmetric functions. Since Sym is contained in the algebra QSym of quasisymmetric functions, this leads us to further generalize immanants with the use of quasisymmetric functions. Since bases of QSym are indexed by integer compositions, we make use of cycle compositions in place of cycle types to define the family of quasi-immanants introduced in this paper. This is achieved through the use of the quasisymmetric power sum bases due to Ballantine et al., and we prove a combinatorial formula for the coefficients arising in an analogue, given by a special case of quasi-immanants associated with quasisymmetric Schur functions, of second immanants.
对于整数划分λ(n)和n×n矩阵A,考虑内嵌imλ (A)的展开式为对称群Sn的不可约字符χλ(ctype(σ))(对于σ的循环类型ctype(σ)∑n),其系数由对称群Sn的不可约字符χλ(ctype(σ))表示。Skandera等人通过将系数χλ(ctype(σ))替换为关于对称函数的代数Sym的不同基间元素的Frobenius态射的原象,引入了对内变量泛化的组合解释。由于Sym包含在准对称函数的代数QSym中,这使得我们可以使用准对称函数进一步推广内嵌式。由于QSym的基是用整数组合来索引的,所以我们用循环组合来代替循环类型来定义本文引入的拟内变量族。这是通过使用由Ballantine等人提出的准对称幂和基来实现的,并且我们证明了由与准对称Schur函数相关的准内变量的特殊情况给出的二次内变量的类比中产生的系数的组合公式。
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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
333
审稿时长
13.8 months
期刊介绍: Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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