Terwilliger algebras of wreath products of one-class association schemes

In this paper, we study the wreath product of one-class association schemes K _n =H(1,n) for n≥2. We show that the d-class association scheme \(K_{n_{1}}\wr K_{n_{2}}\wr \cdots \wr K_{n_{d}}\) formed by taking the wreath product of \(K_{n_{i}}\) (for n _i ≥2) has the triple-regularity property. Then based on this fact, we determine the structure of the Terwilliger algebra of \(K_{n_{1}}\wr K_{n_{2}}\wr \cdots \wr K_{n_{d}}\) by studying its irreducible modules. In particular, we show that every non-primary module of this algebra is 1-dimensional.     

New Effects of an Old Z. Zalcwasser's Theorem

It is well known that the strong (C, 1)-summability of an orthogonal series does not imply its very strong (C, 1)-summability, generally. For a given index-sequence {v _n }, first, Z. Zalcwasser gave an interesting condition implying the strong (C, 1)-summability of these partial sums s _{v n} (x). We show that Zalcwasser's condition on {v _n } holds if and only if the subsequence {v ₂ n} is quasi geometrically increasing. Utilizing this fact and known theorems several strong summability results are presented for given index-sequences {v _n }.     

Ranges of certain functionals in classes of regular functions

LetP _κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M _κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF _α(z)∈P _κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g ^(v?1)(z_?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N _?, on the classP _κ,1(λ,β). On the classesP _κ,n (λ,β),M _κ,n (λ,β,α) one finds the ranges of C_v, v?n, a_m, n?m?2n-2, and ofg(?),F _?(?), 0<¦ξ¦<1, ξ is fixed.     

Cube term blockers without finiteness

We show that an idempotent variety has a d-dimensional cube term if and only if its free algebra on two generators has no d-ary compatible cross. We employ Hall’s Marriage Theorem to show that an idempotent variety \({\mathcal{V}}\) of finite signature whose fundamental operations have arities n ₁, . . . , n _k, has a d-dimensional cube term for some d if and only if it has one of dimension \({1 + \sum_{i=1}^{k} (n_{i} - 1)}\). This upper bound on the dimension of a minimal-dimension cube term for \({\mathcal{V}}\) is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no 2- element semilattice. We prove that the Maltsev condition “existence of a cube term” is join prime in the lattice of idempotent Maltsev conditions.     

Primitivity in the free groups of the variety UmUn

For each m,n >= 0, let G_m,n denote the free group of rank r in the variety U_mU_n. The main results in this paper are: (i) a necessary and sufficient condition for a system of r elements {v₁,…, v_r} to form a basis of G_m,ni (ii) necessary and sufficient conditions for a system of l elements {v₁,…, v_l}, l <= r, to be included in a basis of G_m,0. In particular, (i), (ii) yield corresponding results for the free metabelian group of rank r.     

An analysis of the multiplicity spaces in branching of symplectic groups

Branching of symplectic groups is not multiplicity free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra B{\mathcal{B}}. The algebra B{\mathcal{B}} is a graded algebra whose components encode the multiplicities of irreducible representations of Sp _2n–2 in irreducible representations of Sp _2n . Our first theorem states that the map taking an element of Sp _2n to its principal n × (n + 1) submatrix induces an isomorphism of B{\mathcal{B}} to a different branching algebra B^￠{\mathcal{B}^{\prime}}. The algebra B^￠{\mathcal{B}^{\prime}} encodes multiplicities of irreducible representations of GL _n–1 in certain irreducible representations of GL _n+1. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of Sp _2n to Sp _2n–2 is canonically an irreducible module for the n-fold product of SL ₂. In particular, this induces a canonical decomposition of the multiplicity spaces into one-dimensional spaces, thereby resolving the multiplicities.     

A class of simple Lie algebras attached to unit forms

Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q(x ₁, x ₂,..., x _n) = \({\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}\) and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A _n , and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.     

0-Hecke algebra actions on coinvariants and flags

The 0-Hecke algebra H _n (0) is a deformation of the group algebra of the symmetric group \(\mathfrak{S}_{n}\) . We show that its coinvariant algebra naturally carries the regular representation of H _n (0), giving an analogue of the well-known result for \(\mathfrak{S}_{n}\) by Chevalley–Shephard–Todd. By investigating the action of H _n (0) on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of H _n (0) on the cohomology rings of the Springer fibers, and similarly interpret the (non-commutative) Hall–Littlewood symmetric functions indexed by hook shapes.     

Endomorphism algebras and Igusa–Todorov algebras

Let A be an Artin algebra. If $V\in \operatorname{mod} A$ such that the global dimension of  $\operatorname{End}_{A}V$ is at most 3, then for any ${M\in \operatorname{add}_{A}V}$ , both B and B ^op are 2-Igusa–Todorov algebras, where ${B=\operatorname{End}_{A}M}$ . Let ${P\in \operatorname{mod} A}$ be projective and ${B=\operatorname{End}_{A}P}$ such that the projective dimension of P as a right B-module is at most n(<∞). If A is an m-syzygy-finite algebra (resp. an m-Igusa–Todorov algebra), then B is an (m+n)-syzygy-finite algebra (resp. an (m+n)-Igusa–Todorov algebra); in particular, the finitistic dimension of B is finite in both cases. Some applications of these results are given.     

On twin and anti-twin words in the support of the free Lie algebra

Let ${\mathcal{L}}_{K}(A)$ be the free Lie algebra on a finite alphabet A over a commutative ring K with unity. For a word u in the free monoid A ^? let $\tilde{u}$ denote its reversal. Two words in A ^? are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let l denote the left-normed Lie bracketing and ?? be its adjoint map with respect to the canonical scalar product on the free associative algebra K??A??. Studying the kernel of ?? and using several techniques from combinatorics on words and the shuffle algebra , we show that, when K is of characteristic zero, two words u and v of common length n that lie in the support of ${\mathcal{L}}_{K}(A)$ ??i.e., they are neither powers a ⁿ of letters a??A with exponent n>1 nor palindromes of even length??are twin (resp. anti-twin) if and only if u=v or $u = \tilde{v}$ and n is odd (resp. $u =\tilde{v}$ and n is even).     

On certain forms and quadrics related to symplectic dual polar spaces in characteristic 2

Let V be a 2n-dimensional vector space over a field ${\mathbb {F}}$ and ξ a non-degenerate alternating form defined on V. Let Δ be the building of type C _n formed by the totally ξ-isotropic subspaces of V and, for 1 ≤ k ≤ n, let ${\mathcal {G}_k}$ and Δ _k be the k-grassmannians of PG(V) and Δ, embedded in ${W_k=\wedge^kV}$ and in a subspace ${V_k\subseteq W_k}$ respectively, where ${{\rm dim}(V_k)={2n \choose k} - {2n \choose {k-2}}}$ . This paper is a continuation of Cardinali and Pasini (Des. Codes. Cryptogr., to appear). In Cardinali and Pasini (to appear), focusing on the case of k = n, we considered two forms α and β related to the notion of ‘being at non maximal distance’ in ${\mathcal {G}_n}$ and Δ _n and, under the hypothesis that ${{\rm char}(\mathbb {F}) \neq 2}$ , we studied the subspaces of W _n where α and β coincide or are opposite. In this paper we assume that ${{\rm char}(\mathbb {F}) = 2}$ . We determine which of the quadrics associated to α or β are preserved by the group ${G= {\rm Sp}(2n, \mathbb {F})}$ in its action on W _n and we study the subspace ${\mathcal {D}}$ of W _n formed by vectors v such that α(v, x) = β(v, x) for every ${x \in W_n}$ . Finally, we show how properties of ${\mathcal {D}}$ can be exploited to investigate the poset of G-invariant subspaces of V _k for k = n ? 2i and ${1\leq i \leq \lfloor n/2\rfloor}$ .     

Topological free entropy dimension in unital C-algebras

The notion of topological free entropy dimension of n-tuple of elements in a unital C^∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C^∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C^∗ algebras, including irrational rotation C^∗ algebra, UHF algebra, and minimal tensor product of two reduced C^∗ algebras of free groups.     

Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}      15. Embedding height balanced trees and Fibonacci trees in hypercubes      S. A. Choudum  Indhumathi Raman 《Journal of Applied Mathematics and Computing》2009,30(1-2):39-52 A height balanced tree is a rooted binary tree T in which for every vertex v∈V(T), the difference b T (v) between the heights of the subtrees, rooted at the left and right child of v is at most one. We show that a height-balanced tree T h of height h is a subtree of the hypercube Q h+1 of dimension h+1, if T h contains a path P from its root to a leaf such that $\mathbf{b}_{T_{h}}(v)=1$ , for every non-leaf vertex v in P. A Fibonacci tree $\mathbb{F}_{h}$ is a height balanced tree T h of height h in which $\mathbf{b}_{\mathbb{F}_{h}}(v)=1$ , for every non-leaf vertex. $\mathbb{F}_{h}$ has f(h+2)?1 vertices where f(h+2) denotes the (h+2)th Fibonacci number. Since f(h)～20.694h , it follows that if $\mathbb{F}_{h}$ is a subtree of Q n , then n is at least 0.694(h+2). We prove that $\mathbb{F}_{h}$ is a subtree of the hypercube of dimension approximately 0.75h.      16. Monoids and Groups of <Emphasis Type="Italic">I</Emphasis>-Type      Eric?JespersEmail author  Jan?Okniński 《Algebras and Representation Theory》2005,8(5):709-729 A monoid S generated by {x1,. . .,xn} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaMn of rank n generated by {u1,. . .,un} to S so that for all a∈FaMn one has {v(u1a),. . .,v(una)}={x1v(a),. . .,xnv(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fan and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups. In memory of Paul Wauters Mathematics Subject Classifications (2000) 20F05, 20M05; 16S34, 16S36, 20F16. The authors were supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish–Polish bilateral agreement BIL 01/31, and KBN research grant 2P03A 033 25 (Poland).      17. On the mirabolic Lie algebra \mathfrak{p}_n      A. A. Kirillov 《Functional Analysis and Its Applications》2014,48(2):145-149 We consider the Lie algebra \(\mathfrak{g} = \mathfrak{p}_n \) of (n + 1) × (n + 1) matrices with zeros in the last row. This algebra has received the name of mirabolic; it has many remarkable properties and plays an important role in representation theory. In this paper we study open coadjoint orbits for the corresponding Lie group P n .      18. Quantum affine Gelfand�CTsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces      Aleksander Tsymbaliuk 《Selecta Mathematica, New Series》2010,16(2):173-200 Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL n . We construct the action of the quantum loop algebra Uv(L\mathfraksln){U_v({\bf L}\mathfrak{sl}_n)} in the K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra ü v ([^(\mathfraksl)]n){(\widehat{\mathfrak{sl}}_n)} in the K-theory of the affine version of Laumon spaces.      19. Decomposition of <Emphasis Type="Italic">D</Emphasis>-modules over a hyperplane arrangement in the plane      Tilahun Abebaw  Rikard Bøgvad 《Arkiv f?r Matematik》2010,48(2):211-229       20. Nilmanifolds with a calibrated G2-structure      Diego Conti 《Differential Geometry and its Applications》2011,29(4):493-506 We introduce obstructions to the existence of a calibrated G2-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G2-structure.

Embedding height balanced trees and Fibonacci trees in hypercubes

A height balanced tree is a rooted binary tree T in which for every vertex v∈V(T), the difference b _T (v) between the heights of the subtrees, rooted at the left and right child of v is at most one. We show that a height-balanced tree T _h of height h is a subtree of the hypercube Q _h+1 of dimension h+1, if T _h contains a path P from its root to a leaf such that $\mathbf{b}_{T_{h}}(v)=1$ , for every non-leaf vertex v in P. A Fibonacci tree $\mathbb{F}_{h}$ is a height balanced tree T _h of height h in which $\mathbf{b}_{\mathbb{F}_{h}}(v)=1$ , for every non-leaf vertex. $\mathbb{F}_{h}$ has f(h+2)?1 vertices where f(h+2) denotes the (h+2)th Fibonacci number. Since f(h)～2^0.694h , it follows that if $\mathbb{F}_{h}$ is a subtree of Q _n , then n is at least 0.694(h+2). We prove that $\mathbb{F}_{h}$ is a subtree of the hypercube of dimension approximately 0.75h.     

Monoids and Groups of <Emphasis Type="Italic">I</Emphasis>-Type

A monoid S generated by {x₁,. . .,x_n} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaM_n of rank n generated by {u₁,. . .,u_n} to S so that for all a∈FaM_n one has {v(u₁a),. . .,v(u_na)}={x₁v(a),. . .,x_nv(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fa_n and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups. In memory of Paul Wauters Mathematics Subject Classifications (2000) 20F05, 20M05; 16S34, 16S36, 20F16. The authors were supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish–Polish bilateral agreement BIL 01/31, and KBN research grant 2P03A 033 25 (Poland).     

On the mirabolic Lie algebra \mathfrak{p}_n

We consider the Lie algebra \(\mathfrak{g} = \mathfrak{p}_n \) of (n + 1) × (n + 1) matrices with zeros in the last row. This algebra has received the name of mirabolic; it has many remarkable properties and plays an important role in representation theory. In this paper we study open coadjoint orbits for the corresponding Lie group P _n .     

Quantum affine Gelfand�CTsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL _n . We construct the action of the quantum loop algebra U_v(L\mathfraksl_n){U_v({\bf L}\mathfrak{sl}_n)} in the K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra ü _v ([^(\mathfraksl)]_n){(\widehat{\mathfrak{sl}}_n)} in the K-theory of the affine version of Laumon spaces.     

Nilmanifolds with a calibrated G2-structure

We introduce obstructions to the existence of a calibrated G₂-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G₂-structure.