Category Of Strict Polynomial Functors Research Articles

On the Serre Functor in the Category of Strict Polynomial Functors

AbstractWe study a Serre functor in functor categories related to the category $\mathcal {P}_{d}$ P d of strict polynomial functors over a field of positive characteristic. Our main result shows that the derived category of the category of affine strict polynomial functors in some cases carries the structure of Calabi–Yau category. We also re-obtain the Poincaré duality formulas for Ext groups in $\mathcal {P}_{d}$ P d and construct a certain recollement diagram relating the derived categories of affine and ordinary strict polynomial functors.

Character Polynomials and the Restriction Problem

Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.

Ext-groups in the Category of Strict Polynomial Functors

Abstract The aim of this paper is to study, by using the mathematical tools developed by Chałupnik, Touzé, and Van der Kallen, the effect of the Frobenius twist on $\operatorname{Ext}$-group in the category of strict polynomial functors. As an application, we obtain explicit formulas of cohomology of the orthogonal groups and symplectic ones.

On the Structure of Graded Commutative Exponential functors

Abstract We investigate the structure of graded commutative exponential functors. We give applications of these structure results, including computations of the homology of the symmetric groups and of extensions in the category of strict polynomial functors.

Affine strict polynomial functors and formality

We introduce the notion of an affine strict polynomial functor. We show how this concept helps to understand homological behavior of the operation of Frobenius twist in the category of strict polynomial functors over a field of positive characteristic. We also point out for an analogy between our category and the category of representations of the group of algebraic loops on GLn.

The monoidal structure on strict polynomial functors

The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group Sd which admits a tensor product coming from its Hopf algebra structure. It is classical that there exists a functor F from the category of strict polynomial functors to the category of representations of the symmetric group. Our main result is that this functor F is monoidal. In addition we study the relations under F between projective strict polynomial functors and permutation modules and the link to symmetric functions.

Quantum polynomial functors

We construct a category of quantum polynomial functors which deforms Friedlander and Suslin's category of strict polynomial functors. The main aim of this paper is to develop from first principles the basic structural properties of this category (duality, projective generators, braiding etc.) in analogy with classical strict polynomial functors. We then apply the work of Hashimoto and Hayashi in this context to construct quantum Schur/Weyl functors, and use this to provide new and easy derivations of quantum (GLm,GLn) duality, along with other results in quantum invariant theory.

Poincaré duality for Ext–groups between strict polynomial functors

We study the relation between left and right adjoint functors to the precomposition functor. As a consequence, we obtain various dualities for the Ext–groups in the category of strict polynomial functors.

Derived Kan Extension for Strict Polynomial Functors

We investigate fundamental properties of adjoint functors to the precomposition functor in the category of strict polynomial functors.

Bar complexes and extensions of classical exponential functors

Nous calculons les groupes d’Ext entre foncteurs exponentiels classiques (i.e. puissances symétriques, extérieures ou divisées), et leur précomposition par le foncteur de torsion de Frobenius. Notre méthode repose sur les constructions bar, et relie ces calculs d’Ext avec l’homologie des espaces d’Eilenberg et Mac Lane.

Koszul, Ringel and Serre duality for strict polynomial functors

AbstractThis is a report on recent work of Chałupnik and Touzé. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.

Ringel duality and derivatives of non-additive functors

We prove that Ringel duality in the category of strict polynomial functors can be interpreted as derived functors of non-additive functors (in the sense of Dold and Puppe). We give applications of this fact for both theories.

Troesch complexes and extensions of strict polynomial functors

We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical Ext-computations as well as new results. In particular, we get a cohomological version of the `fundamental theorems' from classical invariant invariant theory for GL_n for n big enough (and we give a conjecture for smaller values of n). We also study the `twisting spectral sequence' E^{s,t}(F,G,r) converging to the extension groups Ext^*(F^{(r)}, G^{(r)}) between the twisted functors F^{(r)} and G^{(r)}. Many classical Ext-computations simply amount to the collapsing of this spectral sequence at the second page (for lacunary reasons), and it is also a convenient tool to study the effect of the Frobenius twist on Ext-groups. We prove many cases of collapsing, and we conjecture collapsing is a general fact.

Extensions of Weyl and Schur functors

We study Ext-groups in the category of strict polynomial functors. The main result is a computation of the Ext-groups between twisted Weyl and Schur functors for a large class of Young diagrams.

Koszul duality and extensions of exponential functors

We study Koszul duality in the category of strict polynomial functors. We compute Koszul duals of various functors and apply these results to the problem of calculating Ext-groups between exponential functors. The main application is a full description of the Ext-groups between twisted exterior and divided powers and between twisted symmetric and divided powers which completes the program started in [V. Franjou, E. Friedlander, A. Scorichenko, A. Suslin, General linear and functor cohomology over finite fields, Ann. of Math. 150 (2) (1999) 663–728].

QUELQUES CALCULS DE COHOMOLOGIE DE COMPOSITIONS DE PUISSANCES SYMÉTRIQUES

ABSTRACT The main purpose of this paper is to compute the Ext-groups and , in the category of functors from finite -vector spaces to all -vect or spaces. This computation gives an approach to the more general computation of . The two main tools are a comparison theorem between Ext-groups in the category and Ext-groups in the category of strict polynomial functors, and the exactness modulo of the post-composition. This means that the homology appearing when applying a functor to a short exact sequence is in the kernel of .