Extensions between exterior powers and between symmetric powers. (Extensions entre puissances extérieures et entre puissances symétriques.) (French) Zbl 0841.55012

Franjou, Vincent

J. Algebra 179, No. 2, 501-522 (1996).

Let \({\mathcal V}_f\) be the category of finite dimensional \(\mathbb{F}_p\)-vector spaces and \({\mathcal V}\) be the category of all \(\mathbb{F}_p\)-vector spaces. This article is a study of the extension groups in the category \({\mathcal F}\) of covariant functors from \({\mathcal V}_f\) into \({\mathcal V}\). These extension groups can be seen as “generic” extensions between representations of the linear groups \(\text{GL}_n\) \((\mathbb{F}_p)\) [N. J. Kuhn, Am. J. Math. 116, 327-360 (1994; Zbl 0813.20049); K-theory 8, 395-428 (1994; Zbl 0830.20065); 9, 273-303 (1995; Zbl 0831.20057).] The author gives some explicit computations of extension groups between exterior powers \(\Lambda^i\) and between symmetric powers \(S_n\) (connectivity of \(\text{Ext}_{\mathcal F} (\Lambda^i, \Lambda^j)\), computation of the algebra \(\text{Ext}_{\mathcal F} (\Lambda^2, \Lambda^2)\), description of \(\text{Ext}_{\mathcal F} (\Lambda^3, \Lambda^n)\), Frobenius map \(\text{Ext}^k_{\mathcal F} (F, S_{np} k) \to \text{Ext}^k_{\mathcal F} (F, S_{np} k + 1)) \).

Reviewer: Y.Felix (Louvain-La-Neuve)

MSC:

55S10	Steenrod algebra
18G15	Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)

Keywords:

extension groups; representations; linear groups

Citations:

Zbl 0813.20049; Zbl 0830.20065; Zbl 0831.20057