Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 038, 21 pages      arXiv:1606.00542
Contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics

Homomorphisms from Specht Modules to Signed Young Permutation Modules

Kay Jin Lim a and Kai Meng Tan b
a) Division of Mathematical Sciences, Nanyang Technological University, SPMS-PAP-03-01, 21 Nanyang Link, 637371 Singapore
b) Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076 Singapore

Received July 14, 2017, in final form April 18, 2018; Published online April 25, 2018

We construct a class $\Theta_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^{\lambda}$ to a signed permutation module $M_{\mathbb{Z}}(\alpha|\beta)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $\phi \in \operatorname{Hom}_{{\mathbb{Z}}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^\lambda, M_{\mathbb{Z}}(\alpha|\beta)\big)$ lies in the $\mathbb{Q}$-span of $\Theta_{\text{sstd}}$, a subset of $\Theta_{\mathscr{R}}$ corresponding to semistandard $\lambda$-tableaux of type $(\alpha|\beta)$. We also study the conditions for which $\Theta^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$ induced by $\Theta_{\mathrm{sstd}}$ - is linearly independent, and show that it is a basis for $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$ when $\mathbb{F}\mathfrak{S}_{n}$ is semisimple.

Key words: symmetric group; Specht module; signed Young permutation module; homomorphism.

pdf (476 kb)   tex (26 kb)


  1. Chuang J., Tan K.M., On certain blocks of Schur algebras, Bull. London Math. Soc. 33 (2001), 157-167.
  2. Danz S., Lim K.J., Signed Young modules and simple Specht modules, Adv. Math. 307 (2017), 369-416, arXiv:1504.02823.
  3. Donkin S., Symmetric and exterior powers, linear source modules and representations of Schur superalgebras, Proc. London Math. Soc. 83 (2001), 647-680.
  4. Du J., Rui H., Quantum Schur superalgebras and Kazhdan-Lusztig combinatorics, J. Pure Appl. Algebra 215 (2011), 2715-2737, arXiv:1010.3800.
  5. Fulton W., Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997.
  6. Hemmer D.J., Irreducible Specht modules are signed Young modules, J. Algebra 305 (2006), 433-441, math.RT/0512469.
  7. James G.D., The representation theory of the symmetric groups, Lecture Notes in Math., Vol. 682, Springer, Berlin, 1978.
  8. James G.D., Kerber A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.
  9. James G.D., Peel M.H., Specht series for skew representations of symmetric groups, J. Algebra 56 (1979), 343-364.
  10. Peel M.H., Hook representations of the symmetric groups, Glasgow Math. J. 12 (1971), 136-149.
  11. Peel M.H., Specht modules and symmetric groups, J. Algebra 36 (1975), 88-97.
  12. Richards M.J., Some decomposition numbers for Hecke algebras of general linear groups, Math. Proc. Cambridge Philos. Soc. 119 (1996), 383-402.
  13. Scopes J., Symmetric group blocks of defect two, Quart. J. Math. Oxford Ser. (2) 46 (1995), 201-234.
  14. Sergeev A.N., Tensor algebra of the identity representation as a module over the Lie superalgebras ${\rm Gl}(n,m)$ and $Q(n)$, Math. USSR Sb. 51 (1985), 419-427.

Previous article  Next article   Contents of Volume 14 (2018)