On the characteristic map of finite unitary groups

In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection allows one to analyze the character theory of the general linear group via symmetric group combinatorics. Using works of Ennola, Kawanaka, Lusztig and Srinivasan, this paper describes the analogous setting for the finite unitary group. In particular, we explain the connection between Deligne–Lusztig theory and Ennola's efforts to generalize Green's work, and from this we deduce various character theoretic results. Applications include calculating certain sums of character degrees, and giving a model of Deligne–Lusztig type for the finite unitary group, which parallels results of Klyachko and Inglis and Saxl for the finite general linear group.     

On the commuting probability and supersolvability of finite groups

For a finite group \(G\) , let \(d(G)\) denote the probability that a randomly chosen pair of elements of \(G\) commute. We prove that if \(d(G)>1/s\) for some integer \(s>1\) and \(G\) splits over an abelian normal nontrivial subgroup \(N\) , then \(G\) has a nontrivial conjugacy class inside \(N\) of size at most \(s-1\) . We also extend two results of Barry, MacHale, and Ní Shé on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if \(d(G)>5/16\) then either \(G\) is supersolvable, or \(G\) isoclinic to \(A_4\) , or \(G/\mathbf{Z}(G)\) is isoclinic to \(A_4\) .     

Loss probability in a finite discrete-time queue in terms of the steady state distribution of an infinite queue

We consider a discrete-time single-server queue with arrivals governed by a stationary Markov chain, where no arrivals are assumed to occur only when the Markov chain is in a particular state. This assumption implies that off-periods in the arrival process are i.i.d. and geometrically distributed. For this queue, we establish the exact relationship between queue length distributions in a finite-buffer queue and the corresponding infinite-buffer queue. With the result, the exact loss probability is obtained in terms of the queue length distribution in the corresponding infinite-buffer queue. Note that this result enables us to compute the loss probability very efficiently, since the queue length distribution in the infinite-buffer queue can be efficiently computed when off-periods are geometrically distributed. This revised version was published online in June 2006 with corrections to the Cover Date.     

再保险与有限时间破产概率

研究保险公司用超额索赔再保险最小化其有限时间破产概率的问题,用鞅方法得到有限时间破产概率的上界以及保险公司的最优再保险自留额.     

On a graph associated to invariant conjugacy classes of finite groups

LetA andG be finite groups of coprime orders such thatA acts by automorphisms onG. We define theA-invariant conjugacy class graph ofG to be the graph having as vertices the noncentralA-invariant conjugacy classes ofG, and two vertices are connected by an edge if their cardinalities are not coprime. We prove that when the graph is disconnected thenG is solvable.     

Some new properties of the Riesz probability distribution

The Riesz probability distribution on any symmetric cone and, in particular, on the cone of positive definite symmetric matrices represents an important generalization of the Wishart and of the matrix gamma distributions containing them as particular examples. The present paper is a continuation of the investigation of the properties of this probability distribution. We first establish a property of invariance of this probability distributions by a subgroup of the orthogonal group. We then show that the Pierce components of a Riesz random variable are independent, and we determine their probability distributions. Some moments and some useful expectations related to the Riesz probability distribution are also calculated. Copyright © 2017 John Wiley & Sons, Ltd.     

The probability of ruin in finite time

The Sparre Andersen model in the collective risk theory is investigated. We obtain the rate of convergence for the ruin probability after thenth payoff in the case where claim sizes are heavy tailed (say, subexponential). Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 304–309, July–September, 1999.     

Character sums associated to finite Coxeter groups

The main result of this paper is a character sum identity for Coxeter arrangements over finite fields which is an analogue of Macdonald's conjecture proved by Opdam.

     

On the pluricanonical map of threefolds of general type

Let be a smooth minimal threefold of general type and let be an integer . Assume that the image of the pluricanonical map of is a curve. Then a simple computation shows that is necessarily or . When with a numerical condition or when , we obtain two inequalities and , where is the irregularity of and is the Euler characteristic of .

     

On upper bounds on stable commutator lengths in mapping class groups

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating curve on an oriented closed surface of genus 2 are not equal to each other.     

On diameter of the commuting graph of a full matrix algebra over a finite field

It is shown that the commuting graph of a matrix algebra over a finite field has diameter at most five if the size of the matrices is not a prime nor a square of a prime. It is further shown that the commuting graph of even-sized matrices over finite field has diameter exactly four. This partially proves a conjecture stated by Akbari, Mohammadian, Radjavi, and Raja [Linear Algebra Appl. 418 (2006) 161–176].     

On e-cuspidal pairs of finite groups of exceptional Lie type

Let G be a simple, simply connected algebraic group of exceptional type defined over ${\mathbb{F}}_q$ with Frobenius endomorphism $F:G\to G$. Let $??q$ be a good prime for G. We determine the number of irreducible Brauer characters in the quasi-isolated ?-blocks of $G^F$. This is done by proving that generalized e-Harish-Chandra theory holds for the Lusztig series associated to quasi-isolated elements of $G^{?F}$.     

On the decomposition map for symmetric groups

Let R ^d be the ℤ-module generated by the irreducible characters of the symmetric group . We determine bases for the kernel of the decomposition map. It is known that R ^d ⊗ _ℤ F is isomorphic to the radical quotient of the Solomon descent algebra when F is a field of characteristic zero. We show that when F has prime characteristic and I _br ^d is the kernel of the decomposition map for prime p then R ^d /I _br ^d ⊗ _ℤ F is isomorphic to the radical quotient of the p-modular Solomon descent algebra. To the memory of Manfred Schocker.