A Bijection for the Alperin Weight Conjecture in <Emphasis Type="Italic">S</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The Alperin weight conjecture states that if G is a finite group and p is a prime, then the number of irreducible Brauer characters of a group G should be equal to the number of conjugacy classes of p-weights of G. This conjecture is known to be true for the symmetric group S _n , however there is no explicit bijection given between the two sets. In this paper we develop an explicit bijection between the p-weights of S _n and a certain set of partitions that is known to have the same cardinality as the irreducible Brauer characters of S _n . We also develop some properties of this bijection, especially in relation to a certain class of partitions whose corresponding Specht modules over fields of characteristic p are known to be irreducible.     

Automorphisms of Coxeter groups of type <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A Coxeter system (W, S) is said to be of type K _n if the associated Coxeter graph Γ_S is complete on n vertices and has only odd edge labels. If W satisfies either of: (1) n = 3; (2) W is rigid; then the automorphism group of W is generated by the inner automorphisms of W and any automorphisms induced by Γ_S. Indeed, Aut(W) is the semidirect product of Inn(W) and the group of diagram automorphisms, and furthermore W is strongly rigid. We also show that if W is a Coxeter group of type K _n then W has exactly one conjugacy class of involutions and hence Aut(W) = Spec(W).     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

Some isomorphism results for Thompson-like groups <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">G</Emphasis>)

The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g ≥ 2 the order of this group is naturally bounded in terms of g due to a Riemann–Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maximal bound Hurwitz translation surfaces. We study for which g there exist Hurwitz translation surfaces of genus g.     

The classification of irreducible admissible mod <Emphasis Type="Italic">p</Emphasis> representations of a <Emphasis Type="Italic">p</Emphasis>-adic GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Let F be a finite extension of ℚ _p . Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over  [`( \mathbbF)]<sub>p</sub>\overline{ \mathbb{F}}_{p} to be supersingular. We then give the classification of irreducible admissible smooth GL <sub>n</sub> (F)-representations over  [`( \mathbbF)]_p\overline{ \mathbb{F}}_{p} in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Continuous <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">p</Emphasis></Subscript>-symmetric distributions

For p > 0, the l _n,p -generalized surface measure on the l _n,p -unit sphere is studied and used for deriving a geometric measure representation for l _n,p -symmetric distributions having a density.     

On the centers of quantum groups of <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-type

Let g be the finite dimensional simple Lie algebra of type An, and let U? = U _q (g,Λ) and U = U _q (g,Q) be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U? for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U? = U _q (g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U? = U _q (g,Λ) and U = U _q (g,Q).     

Structure of the semigroup <Emphasis Type="Italic">OT</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We study the structure of the semigroup OT _n , which is a unique (up to an isomorphism) R-section of the semigroup T _n . For this semigroup, we describe Green relations, determine regular and nilpotent elements, describe maximal nilpotent subsemigroups, and determine the unique irreducible system of generatrices and maximal subsemigroups.     

Finite groups with <Emphasis Type="Italic">S</Emphasis>-supplemented <Emphasis Type="Italic">p</Emphasis>-subgroups

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by B _sG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ B _sG . A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

<Emphasis Type="Italic">n</Emphasis>-Flat and <Emphasis Type="Italic">n</Emphasis>-FP-injective modules

In this paper, we study the existence of the n-flat preenvelope and the n-FP-injective cover. We also characterize n-coherent rings in terms of the n-FP-injective and n-flat modules.     

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

Approximating the Distribution of Customers in <Emphasis Type="Italic">M/E</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Emphasis Type="Italic">/s</Emphasis> Queues

Approximation formulae are suggested for the mean and variance of customers in M/E _n /s queues. It is shown that the distributions can be approximated by using the mean and variance to fit Gamma functions. A brief comment on the more general E _m /E _n /s case is given.     

Congruences Concerning Values of the Modular Functions <Emphasis Type="Italic">j</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The j-function j(z) = q⁻¹+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j_m(ζ) = (j(ζ) – 744)∨T₀(m), where T₀(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL₂(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function. 2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30 The author thanks the National Science Foundation for their generous support.     

Availability Analysis of Repairable <Emphasis Type="Italic">N</Emphasis>-Server <Emphasis Type="Italic">n</Emphasis><Subscript><Emphasis Type="Italic">i</Emphasis></Subscript>-Replicated Systems

We develop an exact closed-form expression for the steady-state availability of a repairable, N-server system in which the ith server contains n _i identical, reconfigurable, breakdown-prone units. Our approach, which follows from the Markov chain balance equations and the recursive properties of Hessenberg matrix determinants, is simpler than previously proposed matrix geometric approaches, and can readily be adapted to the availability analysis of more complicated structures. We illustrate this by computing the steady-state availability of a mixed parallel-serial gracefully degrading replicated system.     

Weierstrass Type Representation of Willmore Surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we reformulate the Euler-Lagrange equations of Willmore surfaces in S^n as the flatness of a family of certain loop algebra-valued 1-forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in S^n and minimal surfaces in constant curvature spaces S^n, R^n, H^n, and prove that some special Willmore surfaces can be derived from minimal surfaces in S^n, R^n, H^n.     

Maximal subgroups of <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">D</Emphasis>) with finite conjugacy classes

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D ^* is an FC-group, then D is a field. Let M be a maximal subgroup of GL _n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M _n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.     

<Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-sequences of algebras with one fundamental operation

We show that if (p₀, p₁, ...) is the p_n-sequence of a nontrivial algebra with one fundamental operation, then p₁ ≥ p₀. Moreover, if , then p₁ > 2p₀. Received April 21, 2003; accepted in final form November 28, 2005.