Fredman’s reciprocity,invariants of abelian groups,and the permanent of the Cayley table

Let C_n{\mathcal{C}}_{n} denote the cyclic group of order n. For G=C_nG={\mathcal{C}}_{n}, we compute the Poincaré series of all C_n{\mathcal{C}}_{n}-isotypic components in (the symmetric tensor exterior algebra of ). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili–Jibladze. Then we consider the Cayley table, , of G and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of equals , where n is the order of G.     

Bundle structure of the Green classes in $M_{n}(\mathbb{K})$

It is shown using the semigroup structure of M_n(\mathbbK)M_{n}(\mathbb{K}) that the Green classes of the multiplicative semigroup M_n(\mathbbK)M_{n}(\mathbb{K}) of linear endomorphisms of an n-dimensional vector space V over \mathbbK\mathbb{K} (=ℝ, or ℂ) are the total spaces of certain fibre bundles having various Grassmann manifolds as the base spaces and the maps x↦R(x), x↦N(x) as the projection maps. In the case of a -class the fibre space is a certain Stiefel manifold and in the case of - and -classes the fibre space is a general linear group. The bundle structures are established by constructing representative coordinate bundles. It is also shown that the bundles having x↦R(x) as the projection map are equivalent to the bundles having x↦N(x) as the projection map.     

Base subsets of symplectic Grassmannians

Let V and V′ be 2n-dimensional vector spaces over fields F and F′. Let also Ω: V× V→ F and Ω′: V′× V′→ F′ be non-degenerate symplectic forms. Denote by Π and Π′ the associated (2n−1)-dimensional projective spaces. The sets of k-dimensional totally isotropic subspaces of Π and Π′ will be denoted by and ${\mathcal G}'_{k}$, respectively. Apartments of the associated buildings intersect and by so-called base subsets. We show that every mapping of to sending base subsets to base subsets is induced by a symplectic embedding of Π to Π′.     

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees . Hultman shows that the complexes and are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.) 10, 37–60, 2004) shows that in fact is subdivided by the order complex of . We introduce the complex of Dowling trees and prove that it is subdivided by the order complex of . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437–468, 2005) shows that, as a simplicial complex, is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that is obtained from by successive coning over certain subcomplexes. It is well known that is shellable, and of the same dimension as . We explicitly and independently calculate how many homology spheres are added in passing from to . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301–336, 2000) shows that is intimely related to the representation theory of the top homology of . Research partially supported by the Swiss National Science Foundation, project PP002-106403/1.     

Derangements and Tensor Powers of Adjoint Modules for \mathfrak{s}\mathfrak{l}_n

We obtain the decomposition of the tensor space as a module for , find an explicit formula for the multiplicities of its irreducible summands, and (when n 2k) describe the centralizer algebra = ( ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of is given by the number of derangements of a set of 2k elements.     

Hecke algebras of finite type are cellular

Let be the one-parameter Hecke algebra associated to a finite Weyl group W, defined over a ground ring in which “bad” primes for W are invertible. Using deep properties of the Kazhdan–Lusztig basis of and Lusztig’s a-function, we show that has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of “Specht modules” for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types A _n and B _n .     

Classification of quasifinite\mathcal{W}_\infty -modules-modules

It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ₁ ^m …D ₁ ^m foru ∈F[Γ] (the group algebra), andm _i≥0 with , whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      12. Weighted Derivations and the <Emphasis Type="Bold">cd</Emphasis>-Index      Duško Jojić 《Discrete and Computational Geometry》2008,39(4):678-689       13. Representation theory of $\mathcal{W}$-algebras      Tomoyuki Arakawa 《Inventiones Mathematicae》2007,169(2):219-320 We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10      14. On some homotopical and homological properties of monoid presentations      Elton Pasku 《Semigroup Forum》2008,76(3):427-468 If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes such that Δ n has dimension n, for every 2≤m≤n, the m-skeleton of Δ n is Δ m , and p m are critical (m+1)-cells with 1≤m≤n−2. For every 2≤m≤n−1, the following is an exact sequence of (ℤS,ℤS)-bimodules where if m=2. We then use these sequences to obtain a free finitely generated bimodule partial resolution of ℤS. Also we show that for groups properties FDT and FHT coincide.      15. Maximal partial packings of $${Z_2^n}$$ with perfect codes      Thomas Westerbäck 《Designs, Codes and Cryptography》2007,42(3):335-355 A maximal partial Hamming packing of is a family of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in . The number of translates of Hamming codes in is the packing number, and a partial Hamming packing is strictly partial if the family does not constitute a partition of . A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of , n = 2 m −1, satisfies . It is also proved that for any n equal to 2 m −1, , there exist maximal strictly partial Hamming packings of with packing numbers n−10,n−9,n−8,...,n−1. This implies that the upper bound is tight for any n = 2 m −1, . All packing numbers for maximal strictly partial Hamming packings of , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.       16. Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities      Evgenii E. Mukhin  Vitaly O. Tarasov  Alexander N. Varchenko 《Functional Analysis and Other Mathematics》2006,1(1):47-69 Let be a space of quasipolynomials of dimension N=N 1+⋅⋅⋅+N n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ i=0 N A N−i (x)∂ x i annihilating V and such that its leading coefficient A 0 is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ i=0 N u i A N−i (∂ u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).      17. On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements      Christos A. Athanasiadis  Eleni Tzanaki 《Journal of Algebraic Combinatorics》2006,23(4):355-375 Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice , spanning a Euclidean space V. Let m be a positive integer and be the arrangement of hyperplanes in V of the form for and . It is known that the number of bounded dominant regions of is equal to the number of facets of the positive part of the generalized cluster complex associated to the pair by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of and conjecture that the corresponding refinement of coincides with the $h$-vector of . We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ, orbits of the action of W on the quotient and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1. 2000 Mathematics Subject Classification Primary—20F55; Secondary—05E99, 20H15      18. Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})      Sihem Ayadi  Slaim Ben Farah 《The Ramanujan Journal》2009,19(2):225-236 We prove real Paley-Wiener type theorems for the Dunkl transform ℱ D on the space of tempered distributions. Let T∈S′(ℝ d ) and Δ κ the Dunkl Laplacian operator. First, we establish that the support of ℱ D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  n→+∞ R −2n Δ κ n T=0 in S′(ℝ d ). Second, we prove that the support of ℱ D (T) is included in ℝ d ∖B(0,M), M>0, if and only if for all R<M, we have lim  n→+∞ R 2n  ℱ D −1(‖y‖−2n ℱ D (T))=0 in S′(ℝ d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.       19. An inequality concerning the elasticity of Krull monoids with divisor class group $\mathbb{Z}_p$      Scott T. Chapman  William W. Smith 《Ricerche di matematica》2007,56(1):107-117 Abstract  Let p be a prime integer and M a Krull monoid with divisor class group . We represent by S the set of nontrivial divisor classes of which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best possible. If M as above has | S| = 3, then it is known that , but for large p, not all the values in this containment set can be realized. For each | S| = 3, we produce a submonoid of such that Keywords: Krull monoid, Block monoid, Elasticity of factorization Mathematics Subject Classification (2000): 20M14, 20D60, 13F05      20. A modified BLM approach to quantum affine {\mathfrak{gl}_n}      Jie Du  Qiang Fu 《Mathematische Zeitschrift》2010,266(4):747-781 We introduce a spanning set of Beilinson–Lusztig–MacPherson type, {A(j, r)} A,j , for affine quantum Schur algebras S\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and construct a linearly independent set {A(j)} A,j for an associated algebra [^(K)]\vartriangle(n){{{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)} . We then establish explicitly some multiplication formulas of simple generators E\vartriangleh,h+1(0){E^\vartriangle_{h,h+1}}(\mathbf{0}) by an arbitrary element A(j) in [^(K)]\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle(n)}} via the corresponding formulas in S\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle(n, r)}} , and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall algebras \mathfrak H\vartriangle(n){{{\boldsymbol{\mathfrak H}_\vartriangle(n)}}} associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for \mathfrak H\vartriangle(n){{\boldsymbol{\mathfrak H}}_\vartriangle}(n) established in Deng and Du (Adv Math 191:276–304, 2005) to derive similar triangular relations for S\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and [^(K)]\vartriangle(n){{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n) . Using these relations, we then show that the subspace \mathfrak A\vartriangle(n){{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)} of [^(K)]\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} spanned by {A(j)} A,j contains the quantum enveloping algebra U\vartriangle(n){{{\mathbf U}_\vartriangle}(n)} of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S(n,r){\boldsymbol{\mathcal S}(n,r)} , the resulting subspace \mathfrak A\vartriangle(n){{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}} is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of \mathfrakgln{\mathfrak{gl}_n} . We conjecture that \mathfrak A\vartriangle(n){{{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}} is a subalgebra of [^(K)]\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} .

Representation theory of $\mathcal{W}$-algebras

We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10     

On some homotopical and homological properties of monoid presentations

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes

Maximal partial packings of $${Z_2^n}$$ with perfect codes

A maximal partial Hamming packing of is a family of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in . The number of translates of Hamming codes in is the packing number, and a partial Hamming packing is strictly partial if the family does not constitute a partition of . A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of , n = 2 ^m −1, satisfies . It is also proved that for any n equal to 2 ^m −1, , there exist maximal strictly partial Hamming packings of with packing numbers n−10,n−9,n−8,...,n−1. This implies that the upper bound is tight for any n = 2 ^m −1, . All packing numbers for maximal strictly partial Hamming packings of , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.      

Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities

Let be a space of quasipolynomials of dimension N=N ₁+⋅⋅⋅+N _n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ _i=0 ^N A _N−i (x)∂ _x ⁱ annihilating V and such that its leading coefficient A ₀ is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ _i=0 ^N u ⁱ A _N−i (∂ _u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).     

On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice , spanning a Euclidean space V. Let m be a positive integer and be the arrangement of hyperplanes in V of the form for and . It is known that the number of bounded dominant regions of is equal to the number of facets of the positive part of the generalized cluster complex associated to the pair by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of and conjecture that the corresponding refinement of coincides with the $h$-vector of . We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ, orbits of the action of W on the quotient and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1. 2000 Mathematics Subject Classification Primary—20F55; Secondary—05E99, 20H15     

Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})

We prove real Paley-Wiener type theorems for the Dunkl transform ℱ _D on the space of tempered distributions. Let T∈S′(ℝ ^d ) and Δ _κ the Dunkl Laplacian operator. First, we establish that the support of ℱ _D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  _n→+∞ R ⁻²ⁿ Δ _κ ⁿ T=0 in S′(ℝ ^d ). Second, we prove that the support of ℱ _D (T) is included in ℝ ^d ∖B(0,M), M>0, if and only if for all R<M, we have lim  _n→+∞ R ²ⁿ  ℱ _D ⁻¹(‖y‖⁻²ⁿ ℱ _D (T))=0 in S′(ℝ ^d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.      

An inequality concerning the elasticity of Krull monoids with divisor class group $\mathbb{Z}_p$

Abstract  Let p be a prime integer and M a Krull monoid with divisor class group . We represent by S the set of nontrivial divisor classes of which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best possible. If M as above has | S| = 3, then it is known that , but for large p, not all the values in this containment set can be realized. For each | S| = 3, we produce a submonoid of such that

A modified BLM approach to quantum affine {\mathfrak{gl}_n}

We introduce a spanning set of Beilinson–Lusztig–MacPherson type, {A(j, r)} _A,j , for affine quantum Schur algebras S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and construct a linearly independent set {A(j)} _A,j for an associated algebra [^(K)]_\vartriangle(n){{{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)} . We then establish explicitly some multiplication formulas of simple generators E^\vartriangle_h,h+1(0){E^\vartriangle_{h,h+1}}(\mathbf{0}) by an arbitrary element A(j) in [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle(n)}} via the corresponding formulas in S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle(n, r)}} , and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall algebras \mathfrak H_\vartriangle(n){{{\boldsymbol{\mathfrak H}_\vartriangle(n)}}} associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for \mathfrak H_\vartriangle(n){{\boldsymbol{\mathfrak H}}_\vartriangle}(n) established in Deng and Du (Adv Math 191:276–304, 2005) to derive similar triangular relations for S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n) . Using these relations, we then show that the subspace \mathfrak A_\vartriangle(n){{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)} of [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} spanned by {A(j)} _A,j contains the quantum enveloping algebra U_\vartriangle(n){{{\mathbf U}_\vartriangle}(n)} of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S(n,r){\boldsymbol{\mathcal S}(n,r)} , the resulting subspace \mathfrak A_\vartriangle(n){{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}} is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of \mathfrakgl_n{\mathfrak{gl}_n} . We conjecture that \mathfrak A_\vartriangle(n){{{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}} is a subalgebra of [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} .