Capelli identities for reductive dual pairs

Analogues of the Capelli identity are given for every irreducible reductive dual pair in the complex symplectic group and the complex orthogonal group. They describe a natural correspondence between the “centers” of the two universal enveloping algebras and an algebra of invariant operators.     

Determinantal ideals and the Capelli identities

Recent work in the formal calculus of variations and also in Riemannian geometry and general relativity have relied upon certain important, previously established properties of determinant ideals in characteristic zero. This note presents direct, elementary proofs of these properties by utilizing the Capelli identities of classical invariant theory.     

Capelli identities for classical Lie algebras

We extend the Capelli identity from the Lie algebra to the other classical Lie algebras and . We employ the theory of reductive dual pairs due to Howe. Received: 12 February 1997 / in revised form: 24 July 1998     

Quantum immanants and higher Capelli identities

We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants. We express them in terms of generatorsE _ij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities. They result in many nontrivial properties of quantum immanants. The author is supported by the International Science Foundation and the Russian Fundamental Research Foundation.     

The exponential growth of codimensions for Capelli identities

By applying the theorem that every positive integer is a sum of four squares, we calculate the exponential growth of the codimensions for the relatively free algebra satisfying Capelli identities. Work partially supported by RFFI grants 96-01-00146 and 98-01-01020. Work partially supported by ISF grant 6629/1. Work partially supported by RFFI grants 96-01-00146 and 96-15-96050.     

Asymptotics of multinomial sums and identities between multi-integrals

We calculate the asymptotics of combinatorial sums ∑ _α f(α)( _α ⁿ ) ^β , whereα = (α ₁, …,α _h ) withα _i =α _j for certaini, j. Hereh is fixed and theα _i ’s are natural numbers. This implies the asymptotics of the correspondingS _n -character degrees ∑_λ f(λ)d _λ ^β . For certain sequences ofS _n characters which involve Young’s rule, the latter asymptotics were obtained earlier [1] by a different method. Equating the two asymptotics, we obtain equations between multi-integrals which involve Gaussian measures. Special cases here give certain extensions of the Mehta integral [5], [6]. Supported by the Weizmann Institute of Science, Rehovot, Israel; by the Institute for Advanced Study, Princeton, New Jersey, USA; NSF grant number DMS 9304580; and by the Centre National de Recherche Scientifique, Lille, France. This work was partially supported by an NSF grant number DMS 94-01197.     

Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities

Let be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence is asymptotic to a function of the form , where and .

     

On P.I. ring and standard identities

An example is given of a ringR (with 1) satisfying the standard identityS ₆[x ₁, ...,x ₆] butM ₂(R), the 2 × 2 matrix ring overR, does not satisfyS ₁₂[x ₁, ...,x ₁₂]. This is in contrast to the caseR=M _n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS _2n [x ₁, ...,x _2n] andM ₂(R) satisfiesS _4n [x ₁, ...,x _n]. Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University of Texas at Austin and the University of Washington at Seattle.     

Capelli elements of the group algebra

Inspired by the Capelli-type identities for group determinants researched by Tôru Umeda, we give Capelli identities for irreducible representations of any finite group, and Capelli elements of the group algebra associated with these identities. These elements construct a basis of the centre of the group algebra.     

On one class of topological *-algebras with standard identities

Let A be a unital semisimple topological nuclear *-algebra over C and let Z be its center. The algebra A is topologically isomorphic to M _n (Z) if and only if A satisfies the standard identity and the maximality condition. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 140–143, January, 2007.     

Asymptotics for the Tukey Median

The asymptotic distribution of the Tukey median has recently been obtained by Nolan in a bivariate setting and by Bai and He in the general multivariate case. To establish their theorem, these authors made a strong symmetry hypothesis on the distribution and, in the case of Bai and He, assumed the existence of a density function with a gradient satisfying a finite-moment condition. This paper extends the above result by deriving the asymptotic distribution of the above location estimate without making any symmetry or differentiability assumption on the distribution.