Strict Polynomial Functors Research Articles

On spectra and affine strict polynomial functors

On spectra and affine strict polynomial functors

On homological properties of strict polynomial functors of degree p

We study the homological algebra in the category Pp of strict polynomial functors of degree p over a field of positive characteristic p. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras of simple functors and Schur functors. We observe that the category Pp has a Kazhdan-Lusztig theory and we show that the DG algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of Pp as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of p-weight 1 in Pe for e>p.

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.

Lifting functors from to

Let (correspondingly ) denote the abelian category of functors (strict polynomial functors in the sense of Friedlander and Suslin) from finite dimensional vector spaces over Fp to vector spaces over Fp . These two categories are related via the exact forgetful functor The category is strongly related to topology and representation theory of symmetric and general linear groups but the homological algebra in is rather mysterious. The category is easier for cohomological calculations. The known calculations are obtained only for functors which belong to the image of ι and are performed using comparison of - and -groups induced by ι. The aim of the following note is to find cohomological conditions which guarantee that a given functor comes from via ι.

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.

RELATING TENSOR STRUCTURES ON REPRESENTATIONS OF GENERAL LINEAR AND SYMMETRIC GROUPS

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 commutative ring 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.

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.

A functorial control of integral torsion in homology

We study the integral torsion of the values of strict polynomial functors defined over the integers. We interpret some classical homological invariants as values of strict polynomial functors, and thus obtain estimates of the integral torsion of these inv

Homogeneous strict polynomial functors as unstable modules

A relation between Schur algebras and the Steenrod algebra is shown in [6] where to each strict polynomial functor the author naturally associates an unstable module. We show that the restriction of Hai's functor to a sub-category of strict polynomial functors of a given degree is fully faithful.

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.

Universal Extension Classes for GL 2

In this note we give a new existence proof for the universal extension classes for G L 2 previously constructed by Friedlander and Suslin via the theory of strict polynomial functors. The key tool in our approach is a calculation of Parker showing that, for suitable choices of coefficient modules, the Lyndon–Hochschild–Serre spectral sequence for S L 2 relative to its first Frobenius kernel stabilizes at the E 2-page. Consequently, we obtain a new proof that if G is an infinitesimal subgroup scheme of G L 2, then the cohomology ring H∙(G,k) of G is a finitely-generated noetherian k-algebra.

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.

Spin polynomial functors and representations of Schur superalgebras

We introduce categories of homogeneous strict polynomial functors, P o l d , k I \mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk } and P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } , defined on vector superspaces over a field k \Bbbk of characteristic not equal 2. These categories are related to polynomial representations of the supergroups G L ( m | n ) GL(m|n) and Q ( n ) Q(n) . In particular, we prove an equivalence between P o l d , k I \mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk } , P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } and the category of finite dimensional supermodules over the Schur superalgebra S ( m | n , d ) \mathcal {S}(m|n,d) , Q ( n , d ) \mathcal {Q}(n,d) respectively provided m , n ≥ d m,n \ge d . We also discuss some aspects of Sergeev duality from the viewpoint of the category P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } .

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.