A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots
Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets J≡(J,⪯). One of the main motivations for the study is an application of matrix representations of posets in representation theory exp...
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2013-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2013/743734 |
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| author | Daniel Simson Katarzyna Zając |
| author_facet | Daniel Simson Katarzyna Zając |
| author_sort | Daniel Simson |
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| description | Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets J≡(J,⪯). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets J such that the symmetric Gram matrix GJ:=(1/2)[CJ+CJtr]∈𝕄J(ℚ) is positive semidefinite, where CJ∈𝕄J(ℤ) is the incidence matrix of J. Following the idea of Drozd mentioned earlier, we associate to J its Coxeter matrix CoxJ:=-CJ·CJ-tr, its Coxeter spectrum speccJ, a Coxeter polynomial coxJ(t)∈ℤ[t], and a Coxeter number cJ. In case GJ is positive semi-definite, we also associate to J a reduced Coxeter number čJ, and the defect homomorphism ∂J:ℤJ→ℤ. In this case, the Coxeter spectrum speccJ is a subset of the unit circle and consists of roots of unity. In case GJ is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets J with the Coxeter spectral properties of a simply laced Euclidean diagram DJ∈{𝔻̃n,𝔼̃6,𝔼̃7,𝔼̃8} associated with J. Our aim of the Coxeter spectral analysis of such posets J is to answer the question when the Coxeter type CtypeJ:=(speccJ,cJ, čJ) of J determines its incidence matrix CJ (and, hence, the poset J) uniquely, up to a ℤ-congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any ℤ-invertible matrix A∈𝕄n(ℤ), there is B∈𝕄n(ℤ) such that Atr=Btr·A·B and B2=E is the identity matrix. |
| format | Article |
| id | doaj-art-ff9e68bcaf0f4ccdb15e5f233086e211 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
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| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ff9e68bcaf0f4ccdb15e5f233086e2112025-02-03T01:10:34ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252013-01-01201310.1155/2013/743734743734A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of RootsDaniel Simson0Katarzyna Zając1Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Ulica Chopina 12/18, 87-100 Toruń, PolandFaculty of Mathematics and Computer Science, Nicolaus Copernicus University, Ulica Chopina 12/18, 87-100 Toruń, PolandFollowing our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets J≡(J,⪯). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets J such that the symmetric Gram matrix GJ:=(1/2)[CJ+CJtr]∈𝕄J(ℚ) is positive semidefinite, where CJ∈𝕄J(ℤ) is the incidence matrix of J. Following the idea of Drozd mentioned earlier, we associate to J its Coxeter matrix CoxJ:=-CJ·CJ-tr, its Coxeter spectrum speccJ, a Coxeter polynomial coxJ(t)∈ℤ[t], and a Coxeter number cJ. In case GJ is positive semi-definite, we also associate to J a reduced Coxeter number čJ, and the defect homomorphism ∂J:ℤJ→ℤ. In this case, the Coxeter spectrum speccJ is a subset of the unit circle and consists of roots of unity. In case GJ is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets J with the Coxeter spectral properties of a simply laced Euclidean diagram DJ∈{𝔻̃n,𝔼̃6,𝔼̃7,𝔼̃8} associated with J. Our aim of the Coxeter spectral analysis of such posets J is to answer the question when the Coxeter type CtypeJ:=(speccJ,cJ, čJ) of J determines its incidence matrix CJ (and, hence, the poset J) uniquely, up to a ℤ-congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any ℤ-invertible matrix A∈𝕄n(ℤ), there is B∈𝕄n(ℤ) such that Atr=Btr·A·B and B2=E is the identity matrix.http://dx.doi.org/10.1155/2013/743734 |
| spellingShingle | Daniel Simson Katarzyna Zając A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots International Journal of Mathematics and Mathematical Sciences |
| title | A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots |
| title_full | A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots |
| title_fullStr | A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots |
| title_full_unstemmed | A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots |
| title_short | A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots |
| title_sort | framework for coxeter spectral classification of finite posets and their mesh geometries of roots |
| url | http://dx.doi.org/10.1155/2013/743734 |
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