Jantzen Sum Formula Research Articles

Positive Jantzen sum formulas for cyclotomic Hecke algebras

This paper proves a “positive” Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A and uses it to obtain new bounds on decomposition numbers. Quite remarkably, our results are proved entirely inside the cyclotomic Hecke algebras. Our positive sum formula shows that, in the Grothendieck group, the Jantzen sum formula can be written as an explicit non-negative linear combination of modules [E^{varvec{nu }}_{f,e}], which are the modular reductions of simple modules of related Hecke algebras in characteristic zero. The coefficient of [E^{varvec{nu }}_{f,e}] in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and by the characteristic of the field. As a consequence we give an explicit upper bound for the decomposition numbers in characteristic p>0 in terms of linear combinations of decomposition numbers of a cyclotomic Hecke algebra at ep^rth roots of unity in characteristic zero, for rge 0. Finally, we prove a new and more elegant “classical” Jantzen sum formula for these algebras.

Coxeter combinatorics for sum formulas in the representation theory of algebraic groups

Let G G be a simple algebraic group over an algebraically closed field F \mathbb {F} of characteristic p ≥ h p \geq h , the Coxeter number of G G . We observe an easy ‘recursion formula’ for computing the Jantzen sum formula of a Weyl module with p p -regular highest weight. We also discuss a ‘duality formula’ that relates the Jantzen sum formula to Andersen’s sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with p p -regular highest weight in terms of the length of the Jantzen filtration of a Weyl module with highest weight in an adjacent alcove.

On the cohomology of line bundles over certain flag schemes II

Over a field K of characteristic p, let Z be the incidence variety in Pd×(Pd)⁎ and let L be the restriction to Z of the line bundle O(−n−d)⊠O(n), where n=p+f with 0≤f≤p−2. We prove that Hd(Z,L) is the simple GLd+1-module corresponding to the partition λf=(p−1+f,p−1,f+1). When f=0, using the first author's description of Hd(Z,L) and Jantzen's sum formula, we obtain as a by-product that the sum of the monomial symmetric functions mλ, for all partitions λ of 2p−1 less than (p−1,p−1,1) in the dominance order, is the alternating sum of the Schur functions Sp−1,p−1−i,1i+1 for i=0,…,p−2.

Šapovalov elements and the Jantzen filtration for contragredient Lie superalgebras: a survey

This is a survey of some recent results on Šapovalov elements and the Jantzen filtration for contragredient Lie superalgebras. The topics covered include the existence and uniqueness of the Šapovalov elements, bounds on the degrees of their coefficients and the behavior of Šapovalov elements when the Borel subalgebra is changed. There is always a unique term whose coefficient has larger degree than any other term. This allows us to define some new highest weight modules. If X is a set of orthogonal isotropic roots and λ ∈ h* is such that λ + ρ is orthogonal to all roots in X, we construct highest weight modules with character ελpx. Here pX is a partition function that counts partitions not involving roots in X. When |X| = 1, these modules are used to give a Jantzen sum formula for Verma modules in which all terms are characters of modules in the category O with positive coefficients.

A Jantzen sum formula for restricted Verma modules over affine Kac–Moody algebras at the critical level

For a restricted Verma module of an affine Kac-Moody algebra at the critical level we describe the Jantzen filtration and give an alternating sum formula which corresponds to the Jantzen sum formula of a baby Verma module over a modular Lie algebra. This also implies a new proof of the linkage principle which was already deduced by Arakawa and Fiebig.

The Brauer algebra and the symplectic Schur algebra

Let k be an algebraically closed field of characteristic p > 0, let m, r be integers with m ≥ 1, r ≥ 0 and m ≥ r and let S 0(2m, r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. We also show that, in general, the block relations of the Brauer algebra and the symplectic and orthogonal Schur algebra are the same. Finally, we deduce from the previous results a new proof of the description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.

Blocks of cyclotomic Hecke algebras

This paper classifies the blocks of the cyclotomic Hecke algebras of type G ( r , 1 , n ) over an arbitrary field. Rather than working with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic Schur algebras. We obtain an explicit description of the blocks by analyzing the combinatorics of ‘Jantzen equivalence.’ We remark that a proof of the classification of the blocks of the cyclotomic Hecke algebras was announced in 1999. Unfortunately, Cox has discovered that this previous proof is incomplete.

A homological interpretation of Jantzen's sum formula

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 GLn (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 GLn 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.

A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras

The Jantzen sum formula for cyclotomic v-Schur algebras yields an identity for some q-analogues of the decomposition matrices of these algebras. We prove a similar identity for matrices of canonical bases of higher-level Fock spaces. We conjecture then that those matrices are actually identical for a suitable choice of parameters. In particular, we conjecture that decomposition matrices of cyclotomic v-Schur algebras are obtained by specializing at q = 1 some transition matrices between the standard basis and the canonical basis of a Fock space.

A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety.

The Jantzen sum formula for cyclotomic 𝑞–Schur algebras

The cyclotomic q q -Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen’s sum formula for the cyclotomic q q -Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.

A q -Analogue of the Jantzen-Schaper Theorem

In this paper we prove an analogue of Jantzen's sum formula for the .1991 Mathematics Subject Classification: 20C32.