Symmetric group representations and Z
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 1-4.

We discuss implications of the following statement about representation theory of symmetric groups: every integer appears infinitely often as an irreducible character evaluation and every nonnegative integer appears infinitely often as a Littlewood–Richardson coefficient and as a Kronecker coefficient.

Nous discutons les implications de l'énoncé suivant en théorie des représentations des groupes symétriques : tout entier apparaît une infinité de fois comme valeur d'un caractère irréductible, et tout entier positif ou nul apparaît une infinité de fois comme coefficient de Littlewood–Richardson et comme coefficient de Kronecker.

Published online:
DOI: 10.1016/j.crma.2017.11.009
Adve, Anshul 1; Yong, Alexander 2

1 University Laboratory High School, Urbana, IL 61801, USA
2 Dept. of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
     author = {Adve, Anshul and Yong, Alexander},
     title = {Symmetric group representations and $ \mathbb{Z}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1--4},
     publisher = {Elsevier},
     volume = {356},
     number = {1},
     year = {2018},
     doi = {10.1016/j.crma.2017.11.009},
     language = {en},
     url = {}
AU  - Adve, Anshul
AU  - Yong, Alexander
TI  - Symmetric group representations and $ \mathbb{Z}$
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 1
EP  - 4
VL  - 356
IS  - 1
PB  - Elsevier
UR  -
DO  - 10.1016/j.crma.2017.11.009
LA  - en
ID  - CRMATH_2018__356_1_1_0
ER  - 
%0 Journal Article
%A Adve, Anshul
%A Yong, Alexander
%T Symmetric group representations and $ \mathbb{Z}$
%J Comptes Rendus. Mathématique
%D 2018
%P 1-4
%V 356
%N 1
%I Elsevier
%R 10.1016/j.crma.2017.11.009
%G en
%F CRMATH_2018__356_1_1_0
Adve, Anshul; Yong, Alexander. Symmetric group representations and $ \mathbb{Z}$. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 1-4. doi : 10.1016/j.crma.2017.11.009.

[1] Bergeron, N.; Sottile, F. Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J., Volume 95 (1998) no. 2, pp. 373-423

[2] Blasiak, J.; Mulmuley, K.D.; Sohoni, M. Geometric complexity theory IV: nonstandard quantum group for the Kronecker problem, Mem. Amer. Math. Soc., Volume 235 (2015) no. 1109

[3] Buch, A.S.; Sottile, F.; Yong, A. Quiver coefficients are Schubert structure constants, Math. Res. Lett., Volume 12 (2005) no. 4, pp. 567-574

[4] Féray, V.; Sniady, P. Asymptotics of characters of symmetric groups related to Stanley character formula, Ann. Math., Volume 173 (2011) no. 2, pp. 887-906

[5] Fulton, W.; Harris, J. Representation Theory, a First Course, Springer-Verlag, 1999

[6] James, G.D. The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, vol. 682, Springer, 1978

[7] James, G.D.; Kerber, A. The Representation Theory of the Symmetric Group, Cambridge University Press, Cambridge, UK, 2009

[8] Knutson, A.; Miller, E.; Shimozono, M. Four positive formulae for type A quiver polynomials, Invent. Math., Volume 166 (2006) no. 2, pp. 229-325

[9] Manivel, L. Symmetric Functions, Schubert Polynomials and Degeneracy Loci, SMF/AMS Texts and Monographs, American Mathematical Society, Providence, RI, USA, 2001 (translated from the 1998 French original by John R. Swallow)

[10] Murnaghan, F.D. The analysis of the Kronecker product of irreducible representations of the symmetric group, Amer. J. Math., Volume 60 (1938) no. 3, pp. 761-784

[11] Narayanan, H. On the complexity of computing Kostka numbers and Littlewood–Richardson coefficients, J. Algebraic Comb., Volume 24 (2006) no. 3, pp. 347-354

[12] Polo, P. Construction of arbitrary Kazhdan–Lusztig polynomials in symmetric groups, Represent. Theory, Volume 3 (1999), pp. 90-104

[13] Roichman, Y. Upper bound on the characters of the symmetric groups, Invent. Math., Volume 125 (1996) no. 3, pp. 451-485

[14] Sagan, B. The Symmetric Group, Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001

[15] Stanley, R.P. Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, UK, 1999

Cited by Sources: