Factorization and Poisson correspondences

The Darboux transformation as an example of an integrable infinite-dimensional Poisson correspondence is discussed in the context of the general factorization problem. Generalizations related to energy-dependent Schrödinger operators and to Kac-Moody algebras are considered. We also present the finite-dimensional reductions of the Darboux transformation to stationary flows.Dedicated to the memory of Ira DorfmanTranslated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 225–245, November, 1995.     

Tableaux and matrix correspondences

The Robinson-Schensted correspondence, a bijection between nonnegative matrices and pair of tableaux, is investigated. The representation, in the tableau, of the dihedral symmetries of the matrix are derived in a simple manner using a shape-preserving anti-isomorphism of the platic monoid. The Robinson-Schensted correspondence is shown to be equivalent to the Hillman-Grassl bijection between reverse plane partitions and tabloids. A generalized insertion method for the Robinson-Schensted correspondence is obtained.     

Fiber-wise birational correspondences

We prove that each fiber-wise birational correspondence between smooth fibrations into (Fano) hypersurfaces, which is biregular on a generic fiber, is a fiber-wise isomorphism. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 120–130, July, 2000.     

Random correspondences and nonlinear equations

The question of existence of random solutions of nonlinear equations involving random correspondences is studied by direct use of the results from the deterministic case without resorting to iterative techniques. Applications to stochastic control and nonlinear random operator equations are given.     

Commuting algebraic correspondences and groups

In this paper, the naturalness of the appearance of the group approach in the study of problems of algebraic commutation is considered. New, previously unknown solutions are obtained. Special attention is paid to an example of commutation obtained from the icosahedral equation. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 662–670, May, 1997. Translated by A. I. Shtern     

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories.We formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

Compositions of theta correspondences

Theta correspondence θ over is established by Howe (J. Amer. Math. Soc. 2 (1989) 535). In He (J. Funct. Anal. 199 (2003) 92), we prove that θ preserves unitarity under certain restrictions, generalizing the result of Li (Invent. Math. 97 (1989) 237). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method (J. Algebra 190 (1997) 518; Representation Theory of Lie Groups, Park City, 1998, pp. 179-238; The Orbit Correspondence for real and complex reductive dual pairs, preprint, 2001), one can compose theta correspondences to obtain unitary representations. We call this process quantum induction.     

On the relative character correspondences

Let π(G,A):Irr_A(G)→Irr(C_G(A)) be the Glauberman–Isaacs correspondence, where G and A are finite groups with coprime orders and A acts on G by automorphisms. Let B be a subgroup of A. In this setting, we give some new conditions for the fixed-point subgroups C_G(A) and C_G(B) such that χπ(G,A) is an irreducible constituent of the restriction of χπ(G,B) to C_G(A) for all χIrr_A(G).     

Sequential convergence via Galois correspondences

Topological sequential spaces are the fixed points of a Galois correspondence between collections of open sets and sequential convergence structures. The same procedure can be followed replacing open sets by other topological concepts, such as closure operators or (ultra)filter convergences. The fixed points of these other Galois correspondences are not topological spaces in general, but they can be embedded into the larger topological classes of pretopological, pseudotopological and convergence spaces. In this paper, we characterize the sequential convergences which are fixed points of these correspondences as well as their restrictions to topological spaces.     

Degree divisibility in character correspondences

Recent results of the author prove the existence of bijections of certain sets of irreducible characters which satisfy many nice properties. In particular, these correspondences imply a strengthened version of the McKay Conjecture for solvable finite groups which takes into account fields of values and Schur indices. In the present paper, we prove that these correspondences have also the additional property that the degree of the corresponding character divides the degree of the original character, and we say something about the quotient as well.     

Four correspondences between graphs and generalized young tableaux

Read's method of counting the number of undirected labeled graphs with a prescribed valency at each labeled node implies that the number of different graphs with a given degree sequence (d₁, d₂, d₃…d_n) is equal to the number of generalized Young tableaux of a certain shape filled with objects of specification (d₁, d₂, d₃…d_n). There are in fact four such results which are applicable to graphs with or without loops and with or without multiple edges. This paper contains four one-one correspondences between the four types of graph and generalized Young tableaux having four different shapes. The correspondences can be considered as combinatorial proofs of four identities of Littlewood.