The Dimension of Skew Shifted Young Diagrams, and Projective Characters of the Infinite Symmetric Group

This article was originally published in Russian in ”Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods. Part 2” (A. M. Vershik, ed.), Zapiski Nauchnyh Seminarov POMI 240 (1997), 115–135 (this text in Russian is available via http://www.pdmi.ras.ru/znsl/1997/v240.html). As it was mentioned in the ”Journal-Ref” field this English translation was published in Journal of Mathematical Sciences (New York) 96 (1999), no. 5, 3517–3530. The dimension of a given skew shifted Young diagram is the number of standard labellings of this diagram. In §1 of this paper, we obtain a formula for the dimension of an arbitrary skew shifted Young diagram. For this purpose, we introduce polynomials P ∗ μ that are factorial analogues of a particular case of Hall–Littlewood polynomials P (·, t) for t = −1 ([3, ch. III, §1]). The definition of the polynomials P ∗ μ is due to A. Yu. Okounkov. As an application of the formula for the dimension of a skew shifted Young diagram, we obtain new proof of the classification of projective characters of the group S(∞). The classic Thoma’s work [12] contains the description of characters (in von Neumann’s sense) of the infinite symmetric group S(∞) that is the inductive limit of the chain of finite symmetric groups S(1) ⊂ S(2) ⊂ . . . (characters in von Neumann’s sense correspond to finite factor–representations). In [5] M. L. Nazarov extended Thoma’s theorem to projective characters of the infinite symmetric group S(∞). In §2 of this paper, we give new proofs of Nazarov’s ([5]) main results. Earlier in [6] and [8] the analogous results in the ordinary (non-projective) case were obtained. In this work we follow the methods of [8]. The author is very grateful to G. I. Olshanski for setting the problem, constant attention to the work and remarks on projects of the manuscript, and to M. L. Nazarov for Remark 1.7 on the formula for the dimension of a skew shifted diagram.