\rm II₁-factor representations of the infinite symmetric inverse semigroup

2018/10/22 by Nessonov, N. I.
#20B30 #20C32 #20M18 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1810.09128

Abstract

Let ℕ be a set of the natural numbers. Symmetric inverse semigroup R_∞ is the semigroup of all infinite 0-1 matrices [ g_ij]_{i,j∈ ℕ} with at most one 1 in each row and each column such that g_ii=1 on the complement of a finite set. The binary operation in R_∞ is the ordinary matrix multiplication. It is clear that infinite symmetric group \mathfrakS_∞ is a subgroup of R_∞. The map ⋆:[ g_ij]↦[ g_ji] is an involution on R_∞. We call a function f on R_∞ positive definite if for all r₁, r₂, …, r_n∈ R_∞ the matrix [ f( r_ir_j^⋆)] is Hermitian and non-negatively definite. A function f said to be indecomposable if the corresponding ⋆-representation π_f is a factor-representation. A class of the R_∞-central functions (characters) is defined by the condition f(rs)=f(sr) for all r,s∈ R_∞. In this paper we classify all factor-representations of R_∞ that correspond to the R_∞-central positive definite functions.

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