Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on the generalized Cartan matrix. We draw special attention to the split case, where the quantum symmetric pair coideal subalgebras are generalized $q$-Onsager algebras.
{"title":"Defining relations for quantum symmetric pair coideals of Kac–Moody type","authors":"Hadewijch De Clercq","doi":"10.4171/JCA/57","DOIUrl":"https://doi.org/10.4171/JCA/57","url":null,"abstract":"Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on the generalized Cartan matrix. We draw special attention to the split case, where the quantum symmetric pair coideal subalgebras are generalized $q$-Onsager algebras.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49243579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is concerned with the constants that appear in Harish-Chandra's character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra's work the only information we have about these constants is that they are uniquely determined by an inductive property. Later Goresky--Kottwitz--MacPherson and Herb gave different formulas for these constants. In this article we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying discrete series character identities proved in a previous paper of the second author. We also introduce a signed convolution of valuations on polyhedral cones in Euclidean space and show that the resulting function is a valuation. This gives a theoretical framework for the valuation appearing in Appendix A of the Goresky--Kottwitz--MacPherson article. In Appendix B we extend the notion of $2$-structures (due to Herb) to pseudo-root systems.
{"title":"A generalization of combinatorial identities for stable discrete series constants","authors":"R. Ehrenborg, S. Morel, Margaret A. Readdy","doi":"10.4171/jca/62","DOIUrl":"https://doi.org/10.4171/jca/62","url":null,"abstract":"This article is concerned with the constants that appear in Harish-Chandra's character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra's work the only information we have about these constants is that they are uniquely determined by an inductive property. Later Goresky--Kottwitz--MacPherson and Herb gave different formulas for these constants. In this article we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying discrete series character identities proved in a previous paper of the second author. We also introduce a signed convolution of valuations on polyhedral cones in Euclidean space and show that the resulting function is a valuation. This gives a theoretical framework for the valuation appearing in Appendix A of the Goresky--Kottwitz--MacPherson article. In Appendix B we extend the notion of $2$-structures (due to Herb) to pseudo-root systems.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47325081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Key map is an important tool in the determination of the Demazure crystals associated to Kac-Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux and Schutzenberger. We show that this procedure is a part of a more general construction holding in the Kac-Moody case that we illustrate in finite types and affine type A. In affine type A, we introduce higher level generalizations of core partitions which notably give interesting analogues of the Young lattice and are expected to parametrize distinguished elements of certain remarkable blocks for Ariki-Koike algebras.
{"title":"Keys and Demazure crystals for Kac–Moody algebras","authors":"N. Jacon, C. Lecouvey","doi":"10.4171/jca/46","DOIUrl":"https://doi.org/10.4171/jca/46","url":null,"abstract":"The Key map is an important tool in the determination of the Demazure crystals associated to Kac-Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux and Schutzenberger. We show that this procedure is a part of a more general construction holding in the Kac-Moody case that we illustrate in finite types and affine type A. In affine type A, we introduce higher level generalizations of core partitions which notably give interesting analogues of the Young lattice and are expected to parametrize distinguished elements of certain remarkable blocks for Ariki-Koike algebras.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47908880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Flat deformations of algebras and functional equations","authors":"B. Feigin, Alexandre Odesski","doi":"10.4171/jca/31","DOIUrl":"https://doi.org/10.4171/jca/31","url":null,"abstract":"","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jca/31","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41822156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We associate a cube complex to any given finitely generated subgroup of a right-angled Coxeter group, called the completion of the subgroup. A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional right-angled Coxeter group. We provide an algorithm that determines whether a given one-ended, 2-dimensional right-angled Coxeter group is isomorphic to some finite-index subgroup of another given right-angled Coxeter group. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund's result that quasiconvex subgroups of right-angled Coxeter groups are separable.
{"title":"Subgroups of right-angled Coxeter groups via Stallings-like techniques","authors":"Pallavi Dani, Ivan Levcovitz","doi":"10.4171/jca/54","DOIUrl":"https://doi.org/10.4171/jca/54","url":null,"abstract":"We associate a cube complex to any given finitely generated subgroup of a right-angled Coxeter group, called the completion of the subgroup. A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional right-angled Coxeter group. We provide an algorithm that determines whether a given one-ended, 2-dimensional right-angled Coxeter group is isomorphic to some finite-index subgroup of another given right-angled Coxeter group. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund's result that quasiconvex subgroups of right-angled Coxeter groups are separable.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48281147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.
{"title":"Cohomology ring of the flag variety vs Chow cohomology ring of the Gelfand–Zetlin toric variety","authors":"Kiumars Kaveh, Elise Villella","doi":"10.4171/jca/56","DOIUrl":"https://doi.org/10.4171/jca/56","url":null,"abstract":"We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44755854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map $w_G:G^drightarrow G$ has a fiber of size at least $rho|G|^d$ for some fixed $rho>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $rho$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $rho$, thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of some independent interest, showing roughly that most elements of large finite permutation groups have large support.
{"title":"Words, permutations, and the nonsolvable length of a finite group","authors":"Alexander Bors, A. Shalev","doi":"10.4171/JCA/51","DOIUrl":"https://doi.org/10.4171/JCA/51","url":null,"abstract":"We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map $w_G:G^drightarrow G$ has a fiber of size at least $rho|G|^d$ for some fixed $rho>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $rho$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $rho$, thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of some independent interest, showing roughly that most elements of large finite permutation groups have large support.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44757942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce polynomially weighted $ell^p$-norms on the bar complex of a finitely generated group. We prove that, for groups of polynomial or exponential growth, the homology of the completed complex does not depend on the value of $p$ in the range $(1,infty)$.
{"title":"Polynomially weighted $ell^p$-completions and group homology","authors":"A. Engel, C. Loeh","doi":"10.4171/jca/40","DOIUrl":"https://doi.org/10.4171/jca/40","url":null,"abstract":"We introduce polynomially weighted $ell^p$-norms on the bar complex of a finitely generated group. We prove that, for groups of polynomial or exponential growth, the homology of the completed complex does not depend on the value of $p$ in the range $(1,infty)$.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43113563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Root graded groups of rank 2","authors":"B. Mühlherr, R. Weiss","doi":"10.4171/JCA/30","DOIUrl":"https://doi.org/10.4171/JCA/30","url":null,"abstract":"","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/30","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49396444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a simple correspondence functor. The method uses the theory of such functors developed in [BT2, BT3], as well as some new ingredients in the theory of finite lattices.
{"title":"The algebra of Boolean matrices, correspondence functors, and simplicity","authors":"S. Bouc, Jacques Th'evenaz","doi":"10.4171/jca/44","DOIUrl":"https://doi.org/10.4171/jca/44","url":null,"abstract":"We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a simple correspondence functor. The method uses the theory of such functors developed in [BT2, BT3], as well as some new ingredients in the theory of finite lattices.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49191939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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