# ALGEBRAIC COMBINATORICS

Linear versus spin: representation theory of the symmetric groups
Algebraic Combinatorics, Volume 3 (2020) no. 1, p. 249-280

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 $\xi$ with analogous normalized linear characters evaluated on the double partition $D\left(\xi \right)$. 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 in terms of counting maps.

Revised : 2019-07-08
Accepted : 2019-08-01
Published online : 2020-02-12
DOI : https://doi.org/10.5802/alco.92
Classification:  20C25,  20C30,  05E05
Keywords: Projective representations of the symmetric groups, linear representations of the symmetric groups, asymptotic representation theory, Stanley character formula
@article{ALCO_2020__3_1_249_0,
author = {Matsumoto, Sho and \'Sniady, Piotr},
title = {Linear versus spin: representation theory of~the symmetric groups},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {3},
number = {1},
year = {2020},
pages = {249-280},
doi = {10.5802/alco.92},
language = {en},
url = {https://alco.centre-mersenne.org/item/ALCO_2020__3_1_249_0}
}

Matsumoto, Sho; Śniady, Piotr. Linear versus spin: representation theory of the symmetric groups. Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 249-280. doi : 10.5802/alco.92. alco.centre-mersenne.org/item/ALCO_2020__3_1_249_0/

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