geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
The representation theory of the symmetric groups.
In characteristic zero, the irreducible representations of the symmetric group are, up to isomorphism, given by the Specht modules labeled by partitions (e.g. Sagan 01, Thm. 2.4.6).
Over the complex numbers:
The dimension of the irrep (Specht module) equals the number of standard Young tableau of shape :
(e.g. Sagan, Thm. 2.6.5)
Moreover, the number of standard Young tableaux of shape is given by the hook length formula
This is due to Frame, Robinson & Thrall 54. Textbook accounts include Stanley 99, Cor. 7.21.6, Sagan 01 Thm. 3.10.2.
Combining (1) with (2) gives the hook length formula for the dimension of the Specht modules
(e.g. James 78, Thm. 20.1)
hook length formula | hook-content formula |
---|---|
number of standard Young tableaux | number of semistandard Young tableaux |
dimension of irreps of Sym(n) | dimension of irreps of SL(n) |
Textbook accounts:
G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, volume 682, Springer 1978 (doi:10.1007/BFb0067708, pdf)
Persi Diaconis, Chapter 7 of: Group Representations in Probability and Statistics, IMS Lecture Notes Monogr. Ser., 11: 198pp. (1988) (jstor:i397389, ISBN: 0940600145, pdf)
William Fulton, Section 7 of: Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997 (doi:10.1017/CBO9780511626241)
Bruce Sagan, The symmetric group, Springer 2001 (doi:10.1007/978-1-4757-6804-6, pdf)
See also:
Notes:
Discussion of characters for the symmetric group that depend only on Cayley distance from the neutral element (“block character”):
From the perspective of the seminormal representation:
On the representation theory of the symmetric group via the seminormal representation:
Part I: Selecta Mathematica, New Series 2, 581-605 (arXiv:math/0503040, doi:10.1007/BF02433451); Part II (incorporates Part I in revised and improved form): Russian version: Записки научных семинаров ПОМИ 307 (2004), 57–98 (Zapiski nauchnyh seminarov POMI 307 (2004), 57–98); English version: Journal of Mathematical Sciences 131 (2005), 5471–5494 (doi:10.1007/s10958-005-0421-7).
In relation to quantum information theory:
Last revised on April 26, 2023 at 15:24:34. See the history of this page for a list of all contributions to it.