The group algebra of the infinite symmetric group

The rational group algebra of the infinite symmetric group is studied using Young diagrams. Maximal and prime ideals are characterized and the maximal condition on ideals is proved. Research supported by the National Research Council of Canada.     

Projective characters of the infinite generalized symmetric group

We consider the infinite generalized symmetric group S(∞)?? _m ^∞ , introduce its covering $\tilde B_m $ , and describe all indecomposable characters on the group $\tilde B_m $ .     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Parity features for classes of the infinite symmetric group

A conjugacy class in the infinite-symmetric group is said to have parity features if no finitary odd permutation is a product of two of its members. The conjugacy classes having parity features are determined. The role played by a property of this kind in determining products of conjugacy classes in any group in which every element is conjugate with its inverse is studied.     

On the Fourier transform on the infinite symmetric group

We present a sketch of the Fourier theory on the in.nite symmetric group S _∞. As a dual space to S _∞, we suggest the space (groupoid) of Young bitableaux B. The Fourier transform of a function on the infinite symmetric group is a martingale with respect to the so-called full Plancherel measure on the groupoid of bitableaux. The Plancherel formula determines an isometry of the space l ²(S _∞, m) of square summable functions on the infinite symmetric group with the counting measure and the space of square integrable functions on the groupoid of bitableaux with the full Plancherel measure. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 61–82.     

The K -functor (Grothendieck group) of the infinite symmetric group

The Grothendieck group K₀() of the group of finite permutations of a countable set is described. All semifinite characters of are described and with their help the cone of representations K ₊ ⁰ () is characterized.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 123, pp. 126–151 (1983).     

Dimensions of skew-shifted young diagrams and projective characters of the infinite symmetric group

We study a factorial of Schur's P-functions. In terms of these functions, we obtain an explicit formula for the dimension of a skew shifted Young diagram. The main application of this formula is a new derivation of the Nazarov's classification of indecomposable projective characters of an infinite symmetric group.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 115–135.Supported by the Soros International Educational Program, grant 2093c.     

A central limit theorem for extremal characters of the infinite symmetric group

The asymptotic behavior of the lengths of the first rows and columns in the random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in n and prove a central limit theorem for their lengths in the case of distinct Thoma parameters. We also prove a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model.     

Automorphisms of a finitary factor power of an infinite symmetric group

We consider a semigroup FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) defined as a finitary factor power of a finitary symmetric group of countable order. It is proved that all automorphisms of FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) are induced by permutations from \mathfrakS( \mathbbN ) \mathfrak{S}\left( \mathbb{N} \right) .     

The small index property and the cofinality of the automorphism group

Within the general model-theoretical framework, we study the small index property and representation of the automorphism group as the union of an increasing chain of proper subsets of a special form.     

On ideals of the group algebra of an infinite symmetric group over a field of characteristic p

We prove that any nonzero ideal of the group algebra of the infinite symmetric group over a field of nonzero characteristic contains skew-symmetric and symmetric elements of sufficiently large order. Using this result, we reduce the question of the classification of the ideals of the group algebra of the infinite symmetric group to the classification of certain subspaces of the tensor square of a finitely generated free associative algebra.     

Groupwise density cannot be much bigger than the unbounded number

We prove that g (the groupwise density number) is smaller or equal to ??⁺, the successor of the minimal cardinality of an unbounded subset of ^ωω. This is true even for the version of ?? for groupwise dense ideals. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)