Skip to main content Accessibility help

A study of the representations supported by the orbit closure of the determinant

  • Shrawan Kumar (a1)


We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated ‘Latin square conjecture’ due to Alon and Tarsi or, more precisely, on the validity of an equivalent ‘column Latin square conjecture’ due to Huang and Rota.



Hide All
[AT92]Alon, N. and Tarsi, M., Colorings and orientations of graphs, Combinatorica 12 (1992), 125134.
[BCI11]Bürgisser, P., Christandl, M. and Ikenmeyer, C., Nonvanishing of Kronecker coefficients for rectangular shapes, Adv. Math. 227 (2011), 20822091.
[BLMW11]Bürgisser, P., Landsberg, J. M., Manivel, L. and Weyman, J., An overview of mathematical issues arising in the geometric complexity theory approach to V PV N P, SIAM J. Comput. 40 (2011), 11791209.
[Dri97]Drisko, A. A., On the number of even and odd Latin squares of order p + 1, Adv. Math. 128 (1997), 2035.
[Gly10]Glynn, D., The conjectures of Alon–Tarsi and Rota in dimension prime minus one, SIAM J. Discrete Math. 24 (2010), 394399.
[GW09]Goodman, R. and Wallach, N., Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol. 255 (Springer, New York, 2009).
[How87]Howe, R., (GLn, GLm)-duality and symmetric plethysm, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), 85109.
[HR94]Huang, R. and Rota, G.-C., On the relations of various conjectures on Latin squares and straightening coefficients, Discrete Math. 128 (1994), 225236.
[Kos76]Kostant, K., On Macdonald’s 𝜂-function formula, the Laplacian and generalized exponents, Adv. Math. 20 (1976), 79212.
[Kum13]Kumar, S., Geometry of orbits of permanents and determinants, Comment. Math. Helv. 88 (2013), 759788.
[MS01]Mulmuley, K. and Sohoni, M., Geometric complexity theory I. An approach to the P vs. NP and related problems, SIAM J. Comput. 31 (2001), 496526.
[MS08]Mulmuley, K. and Sohoni, M., Geometric complexity theory II: Towards explicit obstructions for embeddings among class varieties, SIAM J. Comput. 38 (2008), 11751206.
[Val79]Valiant, L. G., Completeness classes in algebra, in Proceedings of the eleventh annual ACM symposium on theory of computing (Atlanta, GA, 1979) (ACM, New York, 1979), 249261.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed