Let G be a semi-simple simply connected algebraic group over the field C of complex numbers.Let T be a maximal torus of G,and let W be the Weyl group of G with respect to T.Let Z(w,i)be the Bott–Samelson–Demazure–H...Let G be a semi-simple simply connected algebraic group over the field C of complex numbers.Let T be a maximal torus of G,and let W be the Weyl group of G with respect to T.Let Z(w,i)be the Bott–Samelson–Demazure–Hansen variety corresponding to a tuple i associated to a reduced expression of an element w∈W.We prove that for the tuple i associated to any reduced expression of a minuscule Weyl group element w,the anti-canonical line bundle on Z(w,i)is globally generated.As consequence,we prove that Z(w,i)is weak Fano.Assume that G is a simple algebraic group whose type is different from A2.Let S={α1,...,αn}be the set of simple roots.Let w be such that support of w is equal to S.We prove that Z(w,i)is Fano for the tuple i associated to any reduced expression of w if and only if w is a Coxeter element and w^(−1)(Σ_(t=1)^(n)α_(t))∈−S.展开更多
Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an i...Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the 'truncated simple reflections' defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.展开更多
Let w be the element of maximal length in a finite irreducible Coxeter system （W, S）. In the present paper, we get the length of w when （W, S） is of type An, Bn/Cn or Dn.
For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all...For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.展开更多
In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties,...In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap, because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in Coxeter theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in Coxeter theory, which is closely related to conjugacy. We introduce what it means for elements to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.展开更多
The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter gr...The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter group is found. The new base is described by using the notion of cell datum of Graham and Lehrer and the notion of norm.展开更多
An exceptional n-cycle in a Hom-finite triangulated category with Serre functor has been recently introduced by Broomhead,Pauksztello and Ploog.When n=1,it is a spherical object.We explicitly determine all the excepti...An exceptional n-cycle in a Hom-finite triangulated category with Serre functor has been recently introduced by Broomhead,Pauksztello and Ploog.When n=1,it is a spherical object.We explicitly determine all the exceptional cycles in the bounded derived category D^b(kQ)of a finite quiver Q without oriented cycles.In particular,if Q is an Euclidean quiver,then the length type of exceptional cycles in D^b(kQ)is exactly the tubular type of Q;if Q is a Dynkin quiver of type E_m(m=6,7,8),or Q is a wild quiver,then there are no exceptional cycles in D^b(kQ);and if Q is a Dynkin quiver of type An or D_n,then the length of an exceptional cycle in D^b(kQ)is either h or h/2,where h is the Coxeter number of Q.展开更多
Given any Coxeter group, we define rigid reflections and rigid roots using non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, they are related ...Given any Coxeter group, we define rigid reflections and rigid roots using non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, they are related to the rigid representations of the quiver. For a family of rank 3 Coxeter groups, we show that there is a surjective map from the set of reduced positive roots of a rank 2 Kac-Moody algebra onto the set of rigid reflections. We conjecture that this map is bijective.展开更多
Hyperbolic Coxeter polytopes are defined precisely by combinatorial type. Polytopes in hyperbolic n-space with n + p faces that have the combinatorial type of a pyramid over a product of simplices were classified by T...Hyperbolic Coxeter polytopes are defined precisely by combinatorial type. Polytopes in hyperbolic n-space with n + p faces that have the combinatorial type of a pyramid over a product of simplices were classified by Tumarkin for small p. In this article we generalise Tumarkin’s methods and find the remaining hyperbolic Coxeter pyramids.展开更多
This is a pedagogical introduction to the theory of buildings of Jacques Tits and to some applications of this theory.This paper has 4 parts.In the first part we discuss incidence geometry,Coxeter systems and give two...This is a pedagogical introduction to the theory of buildings of Jacques Tits and to some applications of this theory.This paper has 4 parts.In the first part we discuss incidence geometry,Coxeter systems and give two definitions of buildings.We study in the second part the spherical and affine buildings of Chevalley groups.In the third part we deal with Bruhat-Tits theory of reductive groups over local fields.Finally we discuss the construction of the p-adic flag manifolds.展开更多
Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cel...Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cells and two-sided cells of the weighted Coxeter group(W,S,L) that have non-empty intersection with W_J,where the weight function L of(W, S) is in one of the following cases:(i) max{L(s) | s ∈J} < min{L(t)|t∈I};(ii) min{L(s)|s ∈J} ≥L(w_0);(iii) there exists some t ∈ I satisfying L(t) < L(s) for any s ∈I-{t} and L takes a constant value L_J on J with L_J in some subintervals of [1, L(w_0)-1]. The results in the case(iii) are obtained under a certain assumption on(W, W_I).展开更多
The author gives a definition of orbifold Stiefel-Whitney classes of real orbifold vector bundles over special q-CW complexes(i.e.,right-angled Coxeter complexes).Simi-larly to ordinary Stiefel-Whitney classes,orbifol...The author gives a definition of orbifold Stiefel-Whitney classes of real orbifold vector bundles over special q-CW complexes(i.e.,right-angled Coxeter complexes).Simi-larly to ordinary Stiefel-Whitney classes,orbifold Stiefel-Whitney classes here also satisfy the associated axiomatic properties.展开更多
We define and study cocycles on a Coxeter group in each degree generalizing the sign function.When the Coxeter group is a Weyl group,we explain how the degree three cocycle arises naturally from geometric representati...We define and study cocycles on a Coxeter group in each degree generalizing the sign function.When the Coxeter group is a Weyl group,we explain how the degree three cocycle arises naturally from geometric representation theory.展开更多
For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A)are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A ...For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A)are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A⊗K[X]/(X^(N))for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.展开更多
We introduce the mutation game on a directed multigraph,which is dual to Mozes5 numbers game.This new game allows us to create geometric and combinatorial structure that allows generalization of root systems to more g...We introduce the mutation game on a directed multigraph,which is dual to Mozes5 numbers game.This new game allows us to create geometric and combinatorial structure that allows generalization of root systems to more general graphs.We interpret Coxeter-Dynkin diagrams in this multigraph context and exhibit new geometric forms for the associated root systems.展开更多
基金partially supported by a J.C.Bose Fellowship(Grant No.JBR/2023/000003)The second author would like to thank the Infosys Foundation for the partial financial support。
文摘Let G be a semi-simple simply connected algebraic group over the field C of complex numbers.Let T be a maximal torus of G,and let W be the Weyl group of G with respect to T.Let Z(w,i)be the Bott–Samelson–Demazure–Hansen variety corresponding to a tuple i associated to a reduced expression of an element w∈W.We prove that for the tuple i associated to any reduced expression of a minuscule Weyl group element w,the anti-canonical line bundle on Z(w,i)is globally generated.As consequence,we prove that Z(w,i)is weak Fano.Assume that G is a simple algebraic group whose type is different from A2.Let S={α1,...,αn}be the set of simple roots.Let w be such that support of w is equal to S.We prove that Z(w,i)is Fano for the tuple i associated to any reduced expression of w if and only if w is a Coxeter element and w^(−1)(Σ_(t=1)^(n)α_(t))∈−S.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10471071)partially by the Cultivation Fund of the Key Scientific and Technical Innovation Project,Ministry of Education of China 2005.
文摘Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the 'truncated simple reflections' defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.
文摘Let w be the element of maximal length in a finite irreducible Coxeter system （W, S）. In the present paper, we get the length of w when （W, S） is of type An, Bn/Cn or Dn.
文摘For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.
文摘In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap, because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in Coxeter theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in Coxeter theory, which is closely related to conjugacy. We introduce what it means for elements to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.
文摘The concept of norm and cellular algebra are introduced and then the cellular basis is used to replace the Kazhdan-Lusztig basis. So a new base for the center of generic Hecke algebra associated with finite Coxeter group is found. The new base is described by using the notion of cell datum of Graham and Lehrer and the notion of norm.
文摘An exceptional n-cycle in a Hom-finite triangulated category with Serre functor has been recently introduced by Broomhead,Pauksztello and Ploog.When n=1,it is a spherical object.We explicitly determine all the exceptional cycles in the bounded derived category D^b(kQ)of a finite quiver Q without oriented cycles.In particular,if Q is an Euclidean quiver,then the length type of exceptional cycles in D^b(kQ)is exactly the tubular type of Q;if Q is a Dynkin quiver of type E_m(m=6,7,8),or Q is a wild quiver,then there are no exceptional cycles in D^b(kQ);and if Q is a Dynkin quiver of type An or D_n,then the length of an exceptional cycle in D^b(kQ)is either h or h/2,where h is the Coxeter number of Q.
基金supported by a grant from the Simons Foundation (Grant No. 318706)supported by National Science Foundation of USA (Grant No. DMS 1800207)the University of Nebraska-LincolnKorea Institute for Advanced Study
文摘Given any Coxeter group, we define rigid reflections and rigid roots using non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, they are related to the rigid representations of the quiver. For a family of rank 3 Coxeter groups, we show that there is a surjective map from the set of reduced positive roots of a rank 2 Kac-Moody algebra onto the set of rigid reflections. We conjecture that this map is bijective.
文摘Hyperbolic Coxeter polytopes are defined precisely by combinatorial type. Polytopes in hyperbolic n-space with n + p faces that have the combinatorial type of a pyramid over a product of simplices were classified by Tumarkin for small p. In this article we generalise Tumarkin’s methods and find the remaining hyperbolic Coxeter pyramids.
文摘This is a pedagogical introduction to the theory of buildings of Jacques Tits and to some applications of this theory.This paper has 4 parts.In the first part we discuss incidence geometry,Coxeter systems and give two definitions of buildings.We study in the second part the spherical and affine buildings of Chevalley groups.In the third part we deal with Bruhat-Tits theory of reductive groups over local fields.Finally we discuss the construction of the p-adic flag manifolds.
基金supported by National Natural Science Foundation of China (Grant Nos. 11131001 and 11471115)Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeScience and Technology Commission of Shanghai Municipality (Grant No.13dz2260400)
文摘Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cells and two-sided cells of the weighted Coxeter group(W,S,L) that have non-empty intersection with W_J,where the weight function L of(W, S) is in one of the following cases:(i) max{L(s) | s ∈J} < min{L(t)|t∈I};(ii) min{L(s)|s ∈J} ≥L(w_0);(iii) there exists some t ∈ I satisfying L(t) < L(s) for any s ∈I-{t} and L takes a constant value L_J on J with L_J in some subintervals of [1, L(w_0)-1]. The results in the case(iii) are obtained under a certain assumption on(W, W_I).
基金This work was supported by the National Natural Science Foundation of China(No.11971112).
文摘The author gives a definition of orbifold Stiefel-Whitney classes of real orbifold vector bundles over special q-CW complexes(i.e.,right-angled Coxeter complexes).Simi-larly to ordinary Stiefel-Whitney classes,orbifold Stiefel-Whitney classes here also satisfy the associated axiomatic properties.
基金supported by National Natural Science Foundation of China(Grant No. 10731070)the Doctoral Program of Higher Educationthe Fundamental Research Funds for the Central University
文摘In the present paper we determine the representation type of the 0-Hecke algebra of a finite Coxeter group.
文摘We define and study cocycles on a Coxeter group in each degree generalizing the sign function.When the Coxeter group is a Weyl group,we explain how the degree three cocycle arises naturally from geometric representation theory.
文摘For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A)are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A⊗K[X]/(X^(N))for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.
文摘We introduce the mutation game on a directed multigraph,which is dual to Mozes5 numbers game.This new game allows us to create geometric and combinatorial structure that allows generalization of root systems to more general graphs.We interpret Coxeter-Dynkin diagrams in this multigraph context and exhibit new geometric forms for the associated root systems.
