The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Special Linear Group

The minimal polynomials of images of unipotent elements in irreducible rational representations of a special linear group over an algebraically closed field of characteristic p > 2 are found. In particular, we show that the degree of such polynomial is equal to the order of an element provided the highest weight of a representation is in some sense large enough with respect to p.     

On the unitary dual of the classical Lie groups, Representations of

In this paper we prove that a unitary representation of whose infinitesimal character satisfies some regularity condition is infinitesimally isomorphic to an module. This is done using similar techniques as those used by the author in earlier work.

     

version of Hardy's theorem on semisimple Lie groups

We prove an analogue of the version of Hardy's theorem on semisimple Lie groups. The theorem says that on a semisimple Lie group, a function and its Fourier transform cannot decay very rapidly on an average.

     

A Peculiar Property of Exceptional Weyl Groups

Let W be a Weyl group and P W, a parabolic subgroup. In this paper, we give the decomposition of the permutation representation Ind _P ^W 1 into irreducibles for each exceptional W and maximal parabolic P. We find that there is an 'extra' common irreducible component which appears for exceptional groups and not for classical groups. This work is motivated by the study of Prym varieties and integrable systems.     

Supercuspidal representations: An exhaustion theorem

Let be a reductive -adic group. We prove that all supercuspidal representations of arise through Yu's construction subject to certain hypotheses on (depending on ). As a corollary, under the same hypotheses, we see that any supercuspidal representation is compactly induced from a representation of an open subgroup which is compact modulo the center.

     

On Some Applications of Transformation Group Theory

In this article we give several results and problems on the permutation groups, having an interest for cryptography. The abstract of the talk at the international conference Algebra and its applications (Krasnoyarsk, August 5–9, 2002).Mathematics Subject Classifications (2000) 05E20, 20B05, 20F05.     

The Minimal Faithful Permutation Degree for a Direct Product Obeying an Inequality Condition

The minimal faithful permutation degree μ(G) of a finite group G is the least nonnegative integer n such that G embeds in the symmetric group Sym(n). Clearly μ(G × H) ≤ μ(G) + μ(H) for all finite groups G and H. In 1975, Wright ([10 Wright, D. (1975). Degrees of minimal embeddings of some direct products. Amer. J. Math. 97:897–903.[Crossref], [Web of Science ®] , [Google Scholar]]) proved that equality occurs when G and H are nilpotent and exhibited an example of strict inequality where G × H embeds in Sym(15). In 2010 Saunders ([7 Saunders, N. (2010). The minimal degree for a class of finite complex reflection groups. J. Algebra 323:561–573.[Crossref], [Web of Science ®] , [Google Scholar]]) produced an infinite family of examples of permutation groups G and H where μ(G × H) < μ(G) + μ(H), including the example of Wright's as a special case. The smallest groups in Saunders’ class embed in Sym(10). In this article, we prove that 10 is minimal in the sense that μ(G × H) = μ(G) + μ(H) for all groups G and H such that μ(G × H) ≤9.     

Primitive Rank 3 Groups on Symmetric Designs

We determine the symmetric designs which admit a group such that G has a nonabelian socle and is a primitiverank 3 group on points (and blocks).     

A decomposition theorem for polymeasures

We prove that every countably additive polymeasure can be decomposed in a unique way as the sum of a “discrete” polymeasure plus a “continuous” polymeasure. This generalizes a previous result of Saeki.     

A decomposition theorem for neutrices

Neutrices are convex additive subgroups of the nonstandard space R^k, most of them are external sets. Because of the convexity and the invariance under some translations and multiplications, external neutrices are models for orders of magnitude. One dimensional neutrices have been applied to asymptotics, singular perturbations, and statistics. This paper shows that in R^k, with standard k, every neutrix is the direct sum of k neutrices of R. These components may be chosen to be orthogonal.     

A generalized Kakutani-Kodaira theorem

In this note, we present a generalized Kakutani-Kodaira theorem for all locally compact groups.

     

A decomposition for invariant tests of uniformity on the sphere

We introduce a -statistic on which can be based a test for uniformity on the sphere. It is a simple function of the geometric mean of distances between points of the sample and consistent against all alternatives. We show that this type of -statistic, whose kernel is invariant by isometries, can be separated into a set of statistics whose limiting random variables are independent. This decomposition is obtained via the so-called canonical decomposition of a group representation. The distribution of the limiting random variables of the components under the null hypothesis is given. We propose an interpretation of Watson type identities between quadratic functionals of Gaussian processes in the light of this decomposition.

     

Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem

A widely used result of Wedderburn and Artin states that “every left ideal of a ring R is a direct summand of R if and only if R has a unique decomposition as a finite direct product of matrix rings over division rings.” Motivated by this, we call a module M virtually semisimple if every submodule of M is isomorphic to a direct summand of M and M is called completely virtually semisimple if every submodule of M is virtually semisimple. We show that the left R-module R is completely virtually semisimple if and only if R has a unique decomposition as a finite direct product of matrix rings over principal left ideal domains. This shows that R is completely virtually semisimple on both sides if and only if every finitely generated (left and right) R-module is a direct sum of a singular module and a projective virtually semisimple module. The Wedderburn-Artin theorem follows as a corollary from our result.     

Extremal Properties of Bases for Representations of Semisimple Lie Algebras

Let be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of with weight basis . The supporting graph P of is defined to be the directed graph whose vertices are the elements of and whose colored edges describe the supports of the actions of the Chevalley generators on V. Four properties of weight bases are introduced in this setting, and several families of representations are shown to have weight bases which have or are conjectured to have each of the four properties. The basis can be determined to be edge-minimizing (respectively, edge-minimal) by comparing P to the supporting graphs of other weight bases of V. The basis is solitary if it is the only basis (up to scalar changes) which has P as its supporting graph. The basis is a modular lattice basis if P is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of sl(n, ) serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of sp(2n, ) and the irreducible one-dimensional weight space representations of any semisimple Lie algebra, are shown to be solitary and edge-minimal and to have modular lattice supports. Tools developed here are used to construct uniformly the irreducible one-dimensional weight space representations. Similar results for certain irreducible representations of the odd orthogonal Lie algebra o(2n + 1, ), the exceptional Lie algebra G ₂, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.     

A decomposition theorem for Herman maps

In 1980s, Thurston established a topological characterization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed Herman map is developed. It's shown that every Herman map can be decomposed along a stable multicurve into finitely many Siegel maps and Thurston maps, such that the combinations and rational realizations of these resulting maps essentially dominate the original one. This result is motivated by a non-expanding version of McMullen's problem, and Thurston's theory on characterization of rational maps. It enables us to prove a Thurston-type theorem for rational maps with Herman rings.     

A decomposition theorem for maxitive measures

A maxitive measure is the analogue of a finitely additive measure or charge, in which the usual addition is replaced by the supremum operation. In contrast to charges, maxitive measures often have a density. We show that maxitive measures can be decomposed as the supremum of a maxitive measure with density, and a residual maxitive measure that is null on compact sets under specific conditions.