{"title":"Quasi-immanants","authors":"John M. Campbell","doi":"10.1016/j.laa.2025.04.029","DOIUrl":null,"url":null,"abstract":"<div><div>For an integer partition <em>λ</em> of <em>n</em> and an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix <em>A</em>, consider the expansion of the immanant <span><math><msup><mrow><mtext>Imm</mtext></mrow><mrow><mi>λ</mi></mrow></msup><mo>(</mo><mi>A</mi><mo>)</mo></math></span> as a sum indexed by permutations <em>σ</em> of order <em>n</em>, with coefficients given by the irreducible characters <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mi>λ</mi></mrow></msup><mo>(</mo><mtext>ctype</mtext><mo>(</mo><mi>σ</mi><mo>)</mo><mo>)</mo></math></span> of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, for the cycle type <span><math><mtext>ctype</mtext><mo>(</mo><mi>σ</mi><mo>)</mo><mo>⊢</mo><mi>n</mi></math></span> of <em>σ</em>. Skandera et al. have introduced combinatorial interpretations of a generalization of immanants given by replacing the coefficient <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mi>λ</mi></mrow></msup><mo>(</mo><mtext>ctype</mtext><mo>(</mo><mi>σ</mi><mo>)</mo><mo>)</mo></math></span> with preimages with respect to the Frobenius morphism of elements among the distinguished bases of the algebra <span><math><mtext>Sym</mtext></math></span> of symmetric functions. Since <span><math><mtext>Sym</mtext></math></span> is contained in the algebra <span><math><mtext>QSym</mtext></math></span> of quasisymmetric functions, this leads us to further generalize immanants with the use of quasisymmetric functions. Since bases of <span><math><mtext>QSym</mtext></math></span> are indexed by integer compositions, we make use of cycle compositions in place of cycle types to define the family of <em>quasi-immanants</em> 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.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 67-80"},"PeriodicalIF":1.0000,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379525001922","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For an integer partition λ of n and an matrix A, consider the expansion of the immanant as a sum indexed by permutations σ of order n, with coefficients given by the irreducible characters of the symmetric group , for the cycle type of σ. Skandera et al. have introduced combinatorial interpretations of a generalization of immanants given by replacing the coefficient with preimages with respect to the Frobenius morphism of elements among the distinguished bases of the algebra of symmetric functions. Since is contained in the algebra of quasisymmetric functions, this leads us to further generalize immanants with the use of quasisymmetric functions. Since bases of 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.
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.