On simple polynomial -modules
Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for , we consider polynomial representations of for an arbitrary closed reductive subgroup scheme and a maximal torus of in positive characteristic. We give sufficient conditions on making a classification of simple polynomial -modules similar to the case possible and apply this to recover the corresponding result for with a different proof, extending it to symplectic similitude groups, Levi subgroups of and, in a weaker form, to odd orthogonal similitude groups. We also consider orbits of the affine Weyl group and give a condition for equivalence of blocks of polynomial representations for in the case .
When considering rational representations of over an infinite field, one can reduce many questions to polynomial representations. These representations correspond to representations of certain finite-dimensional algebras, the so-called Schur algebras (see for example [G2, Martin1]). Letting be a maximal torus of and be an infinite field of positive characteristic, one can also study the category of -modules to get information on the representation theory of (see [Jantz1, II.9]).
In [DNP2], Doty, Nakano and Peters combined these approaches and considered polynomial representations of for . They show that these representations correspond to representations of the so-called infinitesimal Schur algebras which are subalgebras of the ordinary Schur algebras.
In [Doty1], Doty defined a general notion of polynomial representations and analogues of Schur algebras for algebraic groups contained in . Using his definition, one can study polynomial representations of for other algebraic groups .
If a reductive group admits a graded polynomial representation theory in the sense of [Doty1], it is natural to ask for which character contained in the character group of the simple -module is a polynomial -module. This problem was solved for in [DNP2], but in contrast to most results of that paper, the proof does not carry over to other reductive groups.
Motivated by this problem, we develop a new technique using functions with certain technical properties, see 3.2. The existence of such a function implies a classification of simple polynomial -modules similar to the case (see Theorem 3.6, Corollary 3.9). We give a criterion for the existence of such a function (see Theorem 3.10), allowing us to give a new proof of the result for and extend it to several other cases in a uniform way.
Let be the closure of in the algebraic monoid of -matrices, the set of polynomial weights, i.e. the character monoid of , and the set of -restricted weights. Then our main application can be summarized as follows:
Let be the symplectic dilation group, the torus of diagonal matrices. Then a complete set of representatives for the isomorphism classes of simple polynomial -modules is given by , where and is the character of given by .
Let be the connected component of the odd orthogonal dilation group, the torus of diagonal matrices in . If and all weights of the module are polynomial, then , where and is the character of given by .
Let such that and embed into as a Levi subgroup in the obvious way. Let be the torus of diagonal matrices in , and for , where is the standard basis of . Then a complete set of representatives for the isomorphism classes of simple polynomial -modules is given by , where for .
By studying the intersection of orbits of the affine Weyl group with suitable sets of polynomial weights, we also show that certain shift functors induce Morita equivalences between blocks of polynomial representations of for .
2. Notation and prerequisites
In this section, we fix notation and recall some basic results on polynomial representations. For algebraic groups and , we follow the notation from [Jantz1]. For an introduction to algebraic monoids, we refer the reader to [Ren1].
Let be an infinite perfect field of characteristic and be the monoid scheme of -matrices over . Let and denote by the space generated by all homogeneous polynomials of degree in the coordinate ring of . Then is a graded -bialgebra with comultiplication and counit given by
and each is a subcoalgebra.
As is dense in , the canonical map is injective, so that and each can be viewed as a subcoalgebra of .
Now let be a closed subgroup scheme of and the canonical projection. Set and as well as . Since is a homomorphism of Hopf algebras, is a subbialgebra of and each is a subcoalgebra of . As is finite dimensional, is a finite dimensional associative algebra in a natural way.
For and a maximal torus of , the algebra is the ordinary Schur algebra, while is the infinitesimal Schur algebra defined in [DNP2].
Following [Doty1], we give the following definition.
We say that admits a graded polynomial representation theory if the sum is direct.
We say that a rational -module is a polynomial -module if the corresponding comodule map factors through .
If the comodule map factors through for some , we say that is homogeneous of degree .
Clearly, every -comodule is an -comodule and every -comodule is a -module in a natural way. We record some other properties of polynomial representations from [Doty1, 1.2] in the following proposition.
Suppose admits a graded polynomial representation theory and is a polynomial -module. Then , where each is homogeneous of degree . Furthermore, the category of homogeneous modules of degree is equivalent to the category of -modules.
If contains the center of , then admits a graded polynomial representation theory.
As is a factor bialgebra of , it corresponds to a closed submonoid of . Since is a subbialgebra, is a dense subscheme, so that is the closure of in . Hence polynomial representations of can be regarded as rational representations of the algebraic monoid scheme .
If , a comparison of coordinate rings shows , where with the -th iteration of the Frobenius homomorphism, see [DNP2].
This can fail for general : If is the full orthogonal similitude group of a 2-dimensional vector space with symmetric non-degenerate bilinear form and the torus of diagonal matrices in , then , but is strictly larger than . However, we always have . We do not know whether e.g. for connected .
3. Simple polynomial -modules
In this section, let be a closed connected reductive subgroup scheme containing , be a maximal torus contained in a maximal torus . This is not really a restriction: It is well-known that any affine algebraic group can be embedded into as a closed subgroup for some . If does not contain , we can pass to the group to get a reductive group containing .
Let be the closure of in , be the closure of , be the closure of and be the character monoid of , i.e. the set of polynomial weights, the Weyl group of . Let be the root system of with respect to and a set of simple roots. Denote by the canonical perfect pairing of and the cocharacter group . For , let be the coroot of . Let for all as well as for all .
We want to determine the simple polynomial -modules. The isomorphism classes of simple -modules are parametrized by via , where has highest weight . For convenience and further reference, we collect some basic properties of the from [Jantz1, II.9.6] in the form we need in the following proposition.
Proposition 3.1 ([Jantz1]).
For each , there is a simple -module with . Each weight of satisfies with respect to the standard ordering on .
Each simple -module is isomorphic to exactly one with .
Each is isomorphic to as a -module.
There are isomorphisms of -modules
for all .
If , then , where is the simple -module of highest weight .
As a motivation for our approach, consider the case where and is the maximal torus of diagonal matrices.
According to [DNP2, Section 3], the simple -module is a polynomial -module iff , where
A character belongs to iff for , that is, iff where . Taking this minimum is compatible with multiplication by natural numbers, in particular with multiplication by .
Given characters , we can of course find a permutation such that this minimum is attained at the same coordinate for and , so that
Writing , we have
and we can write
We axiomatize these properties of and the function for general in the following
Suppose there exists a function with the following properties:
Since acts trivially on , and imply that the restriction to of any function as in 3.2 is an isomorphism mapping to .
It will turn out that 3.2 is sufficient for a classification similar to the case . We first show that such a function is essentially unique.
If are functions as in 3.2, there is a permutation of coordinates such that .
Let be the standard basis of and resp. be the preimages of the in for resp. . Then every element of can be written as a linear combination in the with non-negative coefficients. Writing the as such a linear combination and then writing the as such a linear combination in the , we see that the base change matrix for the two bases is a permutation matrix. Thus, there is such that for .
Now let and set . Using and that acts trivially on for the first equality and the fact that is an isomorphism for the second equality, we get
Applying the same arguments to , we get and for . Thus, and for , so that and . This shows , so that , hence . ∎
Suppose is a function as in 3.2 for . Letting be the standard basis of and be the preimages of the in with respect to , we set
Let with . Then there is such that .
We have , where . For all , we have
As , we have
forcing . Thus, and the result follows. ∎
Suppose 3.2 holds for . If , then there is a unique decomposition with . Furthermore, if all weights of the simple -module belong to , then .
Let with .
As , we get that for each , is another decomposition such that . Since generate , 3.5 shows that every decomposition of arises in this fashion. By and since acts trivially on , there is a unique such that . Thus, there is a unique decomposition such that .
We show by contraposition that if all weights of are polynomial, then . Suppose that , so that . Then there is such that . Since , yield . Using and , we see that there is such that
so that by .
Since lifts to a simple -module by 3.1, its character is -invariant, so that is a weight of and is a weight of , so that not all weights of are polynomial. ∎
For , it was shown in the Appendix of [Nak1] that a -module such that all of its weights are polynomial is a polynomial -module. It is not known for which general a similar statement holds. However, if is normal as a variety, we can at least prove this for some simple modules.
Suppose that is a normal variety. If , the simple -module is a polynomial -module.
We have and lifts to by 3.1. Since is normal and is a dominant weight contained in , lifts to by [Doty2, 3.5], so that is a polynomial -module.
As is a -module, is an -module, where is the -th iteration of the Frobenius morphism, and thus a polynomial -module by restriction. Hence is a polynomial -module. ∎
Let be normal and suppose that 3.2 holds for . If , then the simple module is a polynomial -module iff .
Let the derived subgroup of be simply connected and be normal. Suppose 3.2 holds for . Then a system of representatives of isomorphism classes of simple polynomial -modules is given by .
Let . Since the derived subgroup of is simply connected, contains a set of representatives of , so that we have .
By 3.6, all weights of are polynomial iff and by 3.7, is a polynomial -module in this case. Since all weights of polynomial -modules are polynomial and a polynomial module is a simple polynomial -module iff it is simple as a -module, the result follows. ∎
We now give an explicit construction for a function as in 3.2 under certain assumptions.
Suppose there is an action of on such that the canonical projection is -equivariant and there are such that
the sets form a partition of ,
the image of in is contained in the image of and contains the subgroup ,
is a basis of such that each is a linear combination of the with non-negative coefficients.
Then there is a function with the properties of 3.2.
For every , write with . Define
We show that induces a function with the properties of 3.2. For this, we show the following statements .
Let . Since the sets are pairwise disjoint, we have and for every , there is such that . By 3.11 below and , we get for all . As the form a partition of , we get , so that . Consequently, for all .
As , we have
so that for every and . Hence holds.
Clear by definition.
Suppose . We have , so that . Hence . By assumption, the coefficients in the linear combination are non-negative, so that this set is contained in and .
Now let . Since by and since has no negative coordinates and , we have .
Using , we permute the coordinates in each for in such a way that the lowest coordinate for and in each occurs at the same place. Then clearly .
This follows directly from and the definition of .
The statement implies that there is a function such that . Now imply that has the properties of 3.2.
In the situation of 3.10, let and . Then for all .
Let and . Suppose there are such that . Let be the transposition with respect to and . Then . As the restriction of this element is zero in and is torsion-free, , so that . It follows that acts trivially on , a contradiction. ∎
The function defined in the proof of 3.10 provides an easy way to check whether the restriction of a character to is a polynomial weight.
We now apply our results to several examples. The first part of the following corollary is [DNP1, Corollary 3.2]. For the orthogonal and symplectic similitude groups and , we adopt the definition and notation from [Doty1, Sections, 5, 7]. Recall that with suitable choice of defining form, the maximal torus of diagonal matrices in resp. is given by for resp. for .
For the torus of diagonal matrices, a complete set of representatives of simple polynomial -modules is given by , where for .
Let the torus of diagonal matrices in . Then a complete set of representatives of the isomorphism classes of simple polynomial -modules is given by , where and is the character of given by .
Let the torus of diagonal matrices in . If and all weights of the module are polynomial, then , where and is the character of given by .
Let such that and embed into as a Levi subgroup in the obvious way. Let and for . Then a complete set of representatives of isomorphism classes of simple polynomial -modules is given by , where