Certain extensions of completely splittable modules

V. V. Shchigolev

Izvestiya. Mathematics , Tome 68 (2004) no. 4, pp. 833-850.

Résumé

We obtain a formula for $\operatorname{Ext}^1_{K\Sigma_r}(D^\lambda,D^\mu)$ where $K$ is a field of characteristic grater than 2, $\Sigma_r$ is the symmetric group of degree $r$, $D^\lambda$ and $D^\mu$ are simple $K\Sigma_r$-modules, $D^\lambda$ is a completely splittable module, and $\lambda$ does not strictly dominate $\mu$.

Bibliographie

[1] Dzheims G., Teoriya predstavlenii simmetricheskoi gruppy, Mir, M., 1980

[2] Lambek I., Koltsa i moduli, Mir, M., 1971 | MR | Zbl

[3] Schigolev V. V., “Konechnaya baziruemost nekotorykh klassov neprivodimykh predstavlenii simmetricheskikh grupp”, Matem. sb., 194:3 (2003), 149–160 | MR | Zbl

[4] Brundan J., Kleshchev A. S., “On translation functors for general linear and symmetric groups”, Proc. London Math. Soc. (3), 80:1 (2000), 75–106 | DOI | MR | Zbl

[5] Donkin S., “On Schur algebras and related algebras, I”, J. Algebra, 104:2 (1986), 310–328 | DOI | MR | Zbl

[6] Ford B., Kleshchev A. S., “A proof of the Mullineux conjecture”, Math. Z., 226:2 (1997), 267–308 | DOI | MR | Zbl

[7] Green J. A., “Polynomial representations of GL$_n(K)$”, Lecture Notes in Math., 830, Springer-Verlag, Berlin–Heidelberg–N.Y., 1980 | MR

[8] Hemmer D. J., “The $\operatorname{Ext}^1$-quiver for completely splittable representations of the symmetric group”, J. Group Theory, 2001, no. 4, 401–416 | DOI | MR | Zbl

[9] James G. D., Murphy G. E., “The determinant of the Gram matrix for a Specht module”, J. Algebra, 59:1 (1979), 222–235 | DOI | MR | Zbl

[10] James G. D., “On the decomposition matrices of the symmetric groups, III”, J. Algebra, 71:1 (1981), 115–122 | DOI | MR | Zbl

[11] Jantzen J. C., Representations of algebraic groups, Pure and Applied Mathematics, 131, Academic Press, Inc., Boston, 1987 | MR | Zbl

[12] Kleshchev A. S., Nakano D., “On comparing the cohomology of general linear and symmetric groups”, Pacific J. of Math., 201 (2001), 339–355 | MR | Zbl

[13] Kleshchev A. S., Sheth J., “On extensions of simple modules over symmetric and algebraic groups”, J. Algebra, 221:2 (1999), 705–722 | DOI | MR | Zbl

[14] Kleshchev A. S., “Branching rules for modular representations of symmetric groups, I”, J. Algebra, 178:2 (1995), 493–511 | DOI | MR | Zbl

[15] Kleshchev A. S., “Branching rules for modular representations of symmetric groups, II”, J. Reine Angew. Math., 459 (1995), 163–212 | MR | Zbl

[16] Kleshchev A. S., “Completely splittable representations of symmetric groups”, J. Algebra, 181:2 (1996), 584–592 | DOI | MR | Zbl

[17] Mullineux G., “Bijections of $p$-regular partitions and $p$-modular irreducibles of the symmetric groups”, J. London Math. Soc. (2), 20:1 (1979), 60–66 | DOI | MR | Zbl

[18] Shchigolev V. V., “On the stabilization problem for submodules of Specht modules”, J. Algebra, 251:2 (2002), 790–812 | DOI | MR | Zbl