Categories of Bistochastic Measures, and Representations of Some Infinite-Dimensional Groups

Abstract

Based on the inversion pair potentials, the molecular dynamics simulations were performed to study the pressure-induced B1-B2 phase transition in NaCl crystal. Under a transition pressure close to the experimental value, it is found at the temperature of 600 K that there is a transformation from the six-fold coordinated B1 structure to the eight-fold coordinated B2 phase. But at the same temperature, the reversible transformation B2-B1 was presented under zero pressure by the same interionic potentials. These are in good agreement with the experimental observations. The atomistic mechanism of the phase transition was discussed in terms of the configurations under different pressures.

Several Remarks on Groups of Automorphisms of Free Groups

“…The present note shows that a behavior of Aut ∞ is (at least partially) similar to the behavior of infinite-dimensional groups. The proof of Theorem 1.2 given below (2.2) coincides with a proof of [6], Theorem VIII.5.1.…”

“…The phenomenas exist for classical groups over R and over p-adic fields, for symmetric groups, for groups of automorphisms of measure spaces. This was widely explored by G.I.Olshanski in representation theory of infinite-dimensional classical groups (see [12], [11], see also [6]). On recent progress, see, e.g., [9], [7], [8].…”

“…However, this is a general fact for continuous representations of infinite symmetric group (A.Lieberman-G.I.Olshanski, see [10], [6], Section VIII.1, Corollary 5).…”

“…This also determines an action of Γ m on the measure space K m //U by polymorphisms (spreading maps, see [6], Section VIII.4.…”

Tame representations of the group $\operatorname{GL}(\infty,\mathbb{F}_q)$

“…Usually, with every representation of an infinite-dimensional group one can associate a representation of its "mantle" (see [9]), more precisely, of the semigroup of its double cosets (see [2]- [6]). A careful study of this object makes it possible to explicitly construct a discrete imprimitivity system B (see [1]) for each tame representation and to describe the representation in terms of this system.…”