Linear versus spin: representation theory of the symmetric groups

Abstract

We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition ξ with analogous normalized linear characters evaluated on the double partition D(ξ). We also relate some natural filtration on the usual (linear) Kerov-Olshanski algebra of polynomial functions on the set of Young diagrams with its spin counterpart. Finally, we give a spin counterpart to Stanley formula for the characters of the symmetric groups.2010 Mathematics Subject Classification. 20C25 (Primary); 20C30, 05E05 (Secondary).Key words and phrases. projective representations of the symmetric groups, linear representations of the symmetric groups, asymptotic representation theory, Stanley character formula.

Symmetric group characters of almost square shape

“…is a rectangular Young diagram which is almost square. On the other hand, in our recent paper [6] we found an identity which gives a link between the spin characters and their usual (linear) counterparts…”

Random strict partitions and random shifted tableaux

“…The main result of the current section is Theorem 4.2. Our strategy of proof is to use the link between the linear and the spin setup which we explored in [MŚ18]. This section is purely algebraic: all calculations are exact, there are no asymptotic assumptions, there is no randomness, there are no representations and no random Young diagrams.…”

“…Iva06] (see also [MŚ18]), for a fixed odd partition π ∈ OP the corresponding normalized spin character is a function on the set of all strict partitions given by…”

“…These general results about reducible spin representations and shifted Young diagrams are direct analogues of their counterparts for linear representations and usual (non-shifted) Young diagrams. Our strategy will be twofold: to revisit the ideas from the work of the second-named author [Śni06b] which concern the linear representations of the symmetric groups, as well as to use the link between the linear and the spin setup which we explored only recently [MŚ18]. 0.8.…”

Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category

“…Perhaps, we need to take into account more nonlocal effects than just a crossing of two strings. Let us remark that there are analogues of the Kerov conjecture under the Jack-deformed [Las09] and spin settings [Mat18,MŚ20].…”