We prove a strong characteristic-free analogue of the classical adjoint formula s λ s µ f = s λ/µ f in the ring of symmetric functions. This is done by showing that the representative of a suitably chosen functor involving a tensor product is the skew Weyl module. By "strong" we mean that this representative preserves not only Hom groups, but higher Ext groups also-a fact which can be used to compute some homological invariants of Weyl modules for GL n via recursion on degree. We use the following main tools: existence of Weyl filtrations in tensor products of Weyl modules, the Akin-Buchsbaum-Weyman constructions of Weyl modules and certain vanishing properties of Ext groups.
Let V be a Weyl module either for a reductive algebraic group G or for the corresponding quantum group U q . If G is defined over a field of positive characteristic p, respectively if q is a primitive lth root of unity (in an arbitrary field) then V has a Jantzen filtration V = V 0 ⊃ V 1 ⊃ · · · ⊃ V r = 0. The sum of the positive terms in this filtration satisfies a well-known sum formula.If T denotes a tilting module either for G or U q then we can similarly filter the space Hom G (V , T ), respectively Hom U q (V , T ) and there is a sum formula for the positive terms here as well.We give an easy and unified proof of these two (equivalent) sum formulas. Our approach is based on an Euler type identity which we show holds without any restrictions on p or l. In particular, we get rid of previous such restrictions in the tilting module case.
For a split reductive algebraic group, this paper observes a homological interpretation for Weyl module multiplicities in Jantzen's sum formula. This interpretation involves an Euler characteristic χ built from Ext groups between integral Weyl modules. The new interpretation makes transparent For GL n (and conceivable for other classical groups) a certain invariance of Jantzen's sum formula under "Howe duality" in the sense of Adamovich and Rybnikov. For GL n a simple and explicit general formula is derived for χ between an arbitrary pair of integral Weyl modules. In light of Brenti's work on certain R-polynomials, this formula raises interesting questions about the possibility of relating Ext groups between Weyl modules to Kazhdan-Lusztig combinatorics. IntroductionLet G Z be a split and connected reductive algebraic group scheme over Z. For a prime number p, G p will denote the corresponding group scheme over F p , the field of p elements. We will be concerned with (rational) representations of G p , in the course of which we will need to use G Z . We will need some background material on these topics. For all such standard facts see the recent edition of Jantzen's classic text [Jantzen], where one can also find the original references.For a finite dimensional rational representation M of G p , its formal character isW , the Weyl group invariants in the integral group ring of the character group X of a split maximal torus. A central problem is to calculate formal characters of all simple modules, which are in bijective correspondence with their highest weights. To get to the issue of interest in this paper, first let L p (λ) and V p (λ) respectively be the simple module and the Weyl module (i.e., the universal highest weight module) corresponding to a given dominant integral weight λ. Since each of the families {ch(L p (λ)} and {ch(V p (λ))} forms a basis of Z [X] W , and since ch(V p (λ)) is known by Weyl's character formula, the problem 1
For polynomial representations of GL n of a fixed degree, H. Krause defined a new "internal tensor product" using the language of strict polynomial functors. We show that over an arbitrary commutative base ring k, the Schur functor carries this internal tensor product to the usual Kronecker tensor product of symmetric group representations. This is true even at the level of derived categories. The new tensor product is a substantial enrichment of the Kronecker tensor product. E.g. in modular representation theory it brings in homological phenomena not visible on the symmetric group side. We calculate the internal tensor product over any k in several interesting cases involving classical functors and the Weyl functors. We show an application to the Kronecker problem in characteristic zero when one partition has two rows or is a hook.
This paper studies extension groups between certain Weyl modules for the algebraic group GL n over the integers. Main results include: (1) a complete determination of Ext groups between Weyl modules whose highest weights differ by a single root and (2) determination of Ext 1 between an exterior power of the defining representation and any Weyl module. The significance of these results for modular representation theory of GL n is discussed in several remarks. Notably the first result leads to a calculation of Ext groups between neighboring Weyl modules for GL n and also recovers the GL n case of a recent result of Andersen. Some generalities about Ext groups between Weyl modules and a brief overview of known results about these groups are also included.
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