The Birman-Murakami-Wenzl algebras of type E n

The Birman-Murakami-Wenzl algebras (BMW algebras) of type E _n for n = 6; 7; 8 are shown to be semisimple and free over the integral domain \mathbbZ[ d^±1,l^±1,m ]

A Characterization of the Simply-Laced FC-Finite Coxeter Groups

We call an element of a Coxeter group fully covering (or a fully covering element) if its length is equal to the number of the elements it covers in the Bruhat ordering. It is easy to see that the notion of fully covering is a generalization of the notion of a 321-avoiding permutation and that a fully covering element is a fully commutative element. Also, we call a Coxeter group bi-full if its fully commutative elements coincide with its fully covering elements. We show that the bi-full Coxeter groups are the ones of type A_n, D_n, E_n with no restriction on n. In other words, Coxeter groups of type E₉, E₁₀,.... are also bi-full. According to a result of Fan, a Coxeter group is a simply-laced FC-finite Coxeter group if and only if it is a bi-full Coxeter group.AMS Subject Classification: 06A07, 20F55.     

Temperley-Lieb planar algebra modules arising from the ADE planar algebras

A Hilbert module over a planar algebra P is essentially a Hilbert module over a canonically defined algebra spanned by the annular tangles in P. It follows that any planar algebra Q containing P is a module over P, and in particular, any subfactor planar algebra is a module over the Temperley-Lieb planar algebra with the same modulus. We describe a positivity result that allows us to describe irreducible Temperley-Lieb planar algebra modules, and apply the result to decompose the planar algebras determined by the Coxeter graphs A_n (n?3), D_n (n?4), E₆, E₇, and E₈.     

Asymptotically commuting families of operators

We study families of symmetric operators (Q _n) with domains given by the range of self-adjoint contraction semigroups (e ^−tHn ). Assuming the asymptotic commutativity, lim _n [Q _n, e^−tHn]=0, and certain other estimates, we establish the existence and properties of a limiting self-adjoint operatorQ=lim _n Q _n. We apply these results to the study of an elementary supersymmetry algebra. Supported in part by the National Science Foundation under Grant DMS/PHY 88-16214.     

Applications of BGP-reflection functors: isomorphisms of cluster algebras

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.     

Hom-configurations and noncrossing partitions

We study maximal Hom-free sets in the τ[2]-orbit category C(Q) of the bounded derived category for the path algebra associated to a Dynkin quiver Q, where τ denotes the Auslander–Reiten translation and [2] denotes the square of the shift functor. We prove that these sets are in bijection with periodic combinatorial configurations, as introduced by Riedtmann, certain Hom _≤0-configurations, studied by Buan, Reiten and Thomas, and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. Note that Reading has proved that these noncrossing partitions are in bijection with positive clusters in the associated cluster algebra. Finally, we give a definition of mutation of maximal Hom-free sets in C(Q)\mathcal {C}(Q) and prove that the graph of these mutations is connected.     

An Extremal Problem on Degree Sequences of Graphs

 Let G=(I _n ,E) be the graph of the n-dimensional cube. Namely, I _n ={0,1} ⁿ and [x,y]∈E whenever ||x−y||₁=1. For A⊆I _n and x∈A define h _A (x) =#{y∈I _n A|[x,y]∈E}, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A⊆I _n of size 2 ⁿ⁻¹ we have for a universal constant K independent of n. We prove a related lower bound for graphs: Let G=(V,E) be a graph with . Then , where d(x) is the degree of x. Equality occurs for the clique on k vertices. Received: January 7, 2000 RID="*" ID="*" Supported in part by BSF and by the Israeli academy of sciences     

Resolvent Estimates for Operators Belonging to Exponential Classes

For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that s _n (A) = O(exp(-an^α)), where s _n (A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)⁻¹ of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).      

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity

A relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity not of type A _{2 n} , this amounts to the statement that the Poincaré series is the quotient of the characteristic polynomial of the Coxeter element by the characteristic polynomial of the affine Coxeter element of the corresponding root system. We show that this result also follows from the McKay correspondence. Received: Received: 25 October 2001 / Revised version: 19 November 2001     

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type A _n and B _n. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/W _J, where the W _J are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.     

Growth of generalized Temperley–Lieb algebras connected with simple graphs

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , or [(E)\tilde]₇ {\tilde E_7} . An algebra TL_{G, t} T{L_{\Gamma, \tau }} has exponential growth if and only if the graph Γ coincides with none of the graphs A_n {A_n} , D_n {D_n} , E_n {E_n} , [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , and [(E)\tilde]₇ {\tilde E_7} .     

The centralizer algebra of the diagonal action of the group <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(ℂ) in a mixed tensor space

We consider the walled Brauer algebra Br _{k, l}(n) introduced by V. Turaev and K. Koike. We prove that it is a subalgebra of the Brauer algebra and that it is isomorphic, for sufficiently large n ∈ ℕ, to the centralizer algebra of the diagonal action of the group GL_n(ℂ) in a mixed tensor space. We also give the presentation of the algebra Br _{k, l}(n) by generators and relations. For a generic value of the parameter, the algebra is semisimple, and in this case we describe the Bratteli diagram for this family of algebras and give realizations for the irreducible representations. We also give a new, more natural proof of the formulas for the characters of the walled Brauer algebras. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 170–198.     

Quotients of Incidence Algebras and the Euler Characteristic

In this note, we show that, if A ? kQ _A /I _A is a schurian strongly simply connected algebra given by its normed presentation, and Σ is the unique poset whose Hasse quiver coincides with Q _A , then A ? kΣ if and only if I _A has a generating set consisting of exactly χ(Q _A ) elements, where χ(Q _A ) is the Euler characteristic of Q _A . We also prove that a quotient of an incidence algebra A = kΣ/J is strongly simply connected if and only if A is simply connected and kΣ is strongly simply connected.     

The Dual Quiver of the Auslander–Reiten Quiver of Path Algebras

Let Q be a finite quiver of type A _n , n ≥ 1, D _n , n ≥ 4, E ₆, E ₇ and E ₈, σ ∈ Aut(Q), k be an algebraic closed field whose characteristic does not divide the order of σ. In this article, we prove that the dual quiver [(G_Q)\tilde]\widetilde{\Gamma_{Q}} of the Auslander–Reiten quiver Γ _Q of kQ, the Auslander–Reiten quiver of kQ#kás?kQ\#k\langle\sigma\rangle, and the Auslander–Reiten quiver G_[(Q)\tilde]\Gamma_{\widetilde{Q}} of k[(Q)\tilde]k\widetilde{Q}, where [(Q)\tilde]\widetilde{Q} is the dual quiver of Q, are isomorphic.     

Monomials and Temperley–Lieb Algebras

We classify the “fully tight” simply laced Coxeter groups, that is, the ones whoseiji-avoiding Kazhdan–Lusztig basis elements are monomials in the generatorsB_si. We then investigate the basis of the Temperley–Lieb algebra arising from the Kazhdan–Lusztig basis of the associated Hecke algebra, and prove that the basis coincides with the usual (monomial) basis.     

Eine Symmetrieeigenschaft von Solomons Algebra und der höheren Lie-Charaktere

Zusammenfassung  We prove here three results in chain: the result of Section 2 is a symmetry property of the higher Lie characters ofS _n (which are indexed by partitions) : their character table is essentially symmetric, up to well-known factors. This is established using plethystic methods in the algebra of symmetric functions. In Section 3, we show that for any elements ϕ,ωof the Solomon descent algebra ofS _n , one hasc( ϕ)(ω) =c(ω ϕ), wherec is the Solomon mapping from this algebra to the space of central functions onS _n (implicitly extended to its group algebra). We address also the question whether this is true for each finite Coxeter group. Then in the last section, we deduce a new proof of a result of Gessel and the second author that gives the number of permutations with given cycle type and descent set as scalar product of two special characters.     

Generalized Poincaré series for models¶of the braid arrangements

Let be the complexified Coxeter arrangement of hyperplanes of type A _{n −1} (n≥ 3). It is well known that the “minimal” projective De Concini–Procesi model of is isomorphic to the moduli space of stable n plus;1-pointed curves of genus 0. In this paper we study, from the point of view of models of arrangements, the action of the symmetric group Σ _n on the integer cohomology ring of . In fact we find a formula for the generalized Poincaré series which encodes all the information about this representation of Σ _n . This formula, which is obtained by using the elementary combinatorial properties of a ℤ-basis of and turns out to be very direct, should be compared with a more general result due to Getzler (see [5]). Received: 24 November 1997 / Revised version: 23 April 1998     

A superadditivity and submultiplicativity property for cardinalities of sumsets

For finite sets of integers A ₁,…,A _n we study the cardinality of the n-fold sumset A ₁+…+ A _n compared to those of (n−1)-fold sumsets A ₁+…+A _i−1+A _i+1+…+A _n . We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets.     

Graded automorphism group of TKK Lie algebra over semilattice

Every extended affine Lie algebra of type A ₁ and nullity ν with extended affine root system R(A ₁, S), where S is a semilattice in ℝ ^ν , can be constructed from a TKK Lie algebra T (J (S)) which is obtained from the Jordan algebra J (S) by the so-called Tits-Kantor-Koecher construction. In this article we consider the ℤ ⁿ -graded automorphism group of the TKK Lie algebra T (J (S)), where S is the “smallest” semilattice in Euclidean space ℝ ⁿ .