Endomorphism Algebras of Some Modules for Schur Algebras and Representation Dimension
Stephen Donkin, Haralampos Geranios

TL;DR
This paper investigates the representation dimension of Schur algebras, providing bounds in both ordinary and quantum cases, with implications for understanding their module categories.
Contribution
It establishes lower bounds for the representation dimension of Schur algebras in positive characteristic and upper bounds in the quantum case over characteristic zero fields.
Findings
Lower bounds for representation dimension in positive characteristic
Upper bounds in the quantum case over characteristic zero
Insights into module category complexity of Schur algebras
Abstract
We consider the representation dimension, for fixed , of ordinary and quantised Schur algebras over a field . For of positive characteristic we give a lower bound valid for all . We also give an upper bound in the quantum case, when has characteristic .
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Advanced Combinatorial Mathematics
Endomorphism Algebras of Some Modules for Schur Algebras
and Representation Dimension
Stephen Donkin and Haralampos Geranios
Department of Mathematics, University of York, York YO10 5DD
11 February 2013
Abstract
We consider the representation dimension, for fixed , of ordinary and quantised Schur algebras over a field . For of positive characteristic we give a lower bound valid for all . We also give an upper bound in the quantum case, when has characteristic [math].
1 Introduction
In a recent paper [14], Miemietz and Oppermann describe a lower bound for the representation dimension of Schur algebras over a field of characteristic . This bound is stated under the condition . We here give a bound valid for all . Our main interest is in the classical case so the main body of the text is expressed in the classical context. However, we point out that our results are valid also in the quantum case in positive characteristic. We also point out that for fixed there is an upper bound of the representation dimension of the -Schur algebras valid for all in the characteristic [math] case.
There are certain similarities between our approach and that of Miemietz and Oppermann. In particular we follow their method of finding modules with endomorphism algebra isomorphic to a truncated polynomial algebra (we work with injective modules and they with projective modules). However, in detail our techniques are very different and our main technique is to compare the representation theory of the general linear group with that of its infinitesimal subgroups. Moreover, we avoid certain problems with the argument of Miemietz and Oppermann (see the Remark of Section 3 for details). However, as in [14], we are still reliant on the work of Bergh, [2], Oppermann, [15] and Rouquier, [17, 18], relating representation dimension and certain endomorphism algebras via derived categories.
In Section 2 we deal with the preliminary background material and notation. In Section 3 we consider certain polynomial injective modules and their endomoprhism algebras. In Section 4 we consider further the endomorphism algebras of these modules and deduce that, for fixed , the representation dimension of grows with . For the precise result, see Corollary 4.4.
In the final section we work over a quantum general linear group at a non-zero parameter . In particular we demonstrate a certain dichotomy. We first consider the characteristic [math] case and show, Proposition 5.2, that for (and ) fixed the representation dimension of the Schur algebra (as varies) is bounded above. We then consider the case in which has positive characteristic and show, Theorem 5.3, that if is a root of unity then, as in the classical case, the representation dimension of grows with .
2 Preliminaries
We start with some standard combinatorics associated with the representation theory of general linear groups. We fix a positive integer . We set . There is a natural partial order on . For , we write if for and . We shall use the standard -basis of , where (with in the th position). We shall also need the specific elements and . The symmetric group acts naturally on . We write for the longest element of , i.e., the element such that , for .
We write for the set of dominant weights, i.e., the set of such that . We write for the set of polynomial weights, i.e., the set of with all . We set , the set of polynomial dominant weights. For we define its degree and if , define its breadth . For a positive integer we write for the set of all such that . An element of is called a partition of into at most parts. For an integer we say that is column -regular if for and .
We form the integral group ring of . This has -basis of formal exponentials , , which multiply according to the rule . The action of extends to an action on by ring automorphisms. For we write for the orbit sum .
We fix an algebraically closed field and a positive integer . We write or for the general linear group . We write for the coordinate algebra of . For let denote the corresponding coordinate function, i.e., the function taking to its entry. We set . Then is a free polynomial algebra in generators . Moreover has a -space decomposition , where is the span of the all monomials .
We write or for the subgroup of consisting of all invertible diagonal matrices. For we also denote by the multiplicative character of defined by , where denotes the -entry of the diagonal matrix , and in this way identify with the character group of . For we write for the one dimensional -module on which acts as multiplication by . The modules , , form a complete set of pairwise non-isomorphic rational -modules. We write , or (resp. or ) for the subgroup of consisting of all lower (resp. upper) triangular invertible matrices. For , the action of extends uniquely to a module action of (resp. ) on and the modules , , form a complete set of pairwise non-isomorphic irreducible rational -modules (resp. -modules).
For we write for the induced module . The dual of a finite dimensional rational -module will be denoted . We also have the Weyl module , for . A good filtration (resp. Weyl filtration) of a rational -module is a filtration such that for each the section is either [math] or isomorphic to (resp. ) for some . If has a good filtration (resp. Weyl filtration) then, for , the number of sections in the filtration isomorphic to (resp. ) is independent of the choice of good filtration (resp. Weyl filtration) and will be denoted (resp. ).
For a rational -module and we have the weight space . A rational -module decomposes as a direct sum of its weight spaces: . To a finite dimensional rational -module we attach its character
[TABLE]
For the character is given by Weyl’s character formula, [[13], II, 5.10 Proposition]. For each there is an irreducible rational -module , with unique highest weight occurring with multiplicity one, and the modules , , form a complete set of pairwise non-isomorphic irreducible rational -modules. For we write for the th Steinberg module . We also write simply for . We shall write for the determinant module, i.e, the one dimensional -module on which acts as multiplication by . For a finite dimensional -module and we write for the -fold tensor product . We write for the dual module . In particular we have , for any integer . We write for the space of column vectors of length . Then is a -module (the natural module) with action by matrix multiplication. For the exterior power is irreducible of highest weight (where occurs times). So we have and in particular . For a finite dimensional rational -module and we write for the multiplicity of as a composition factor of .
A finite dimensional rational module which has a filtration by induced modules and a filtration by Weyl modules is called a tilting module. For each there exists an indecomposable tilting module which has a weight occurring with multiplicity one and all other weights smaller than . Moreover, the modules , , form a complete set of pairwise non-isomorphic indecomposable tilting modules.
We write for the space of all -valued functions on . Let be a finite dimensional rational module with basis . The coefficient functions , , are defined by the equations
[TABLE]
for , . The coefficient space is the -span of the coefficient functions , .
A -module is called polynomial if and polynomial of degree if . A polynomial -module has a unique module decomposition
[TABLE]
where is polynomial of degree . The coordinate algebra has a natural Hopf algebra structure and each space is a subcoalgebra. The dual space has a natural algebra structure. The algebras are called Schur algebras. The category of polynomial modules of degree is naturally equivalent to the category of -modules.
The modules , , form a complete set of pairwise non-isomorphic irreducible polynomial modules. For we write for the injective hull of in the category of polynomial modules. Then is finite dimensional, indeed it is may be identified with the injective hull of as a module for the Schur algebra , where . For further details of the category of polynomial modules in the spirit of this paper see for example, [10], [7],[9]. We shall use, without further reference the fact that a rational -module is polynomial if and only if all of its composition factors are of the form , - this follows from [4], Section 1, (1) for example. We shall call a -module polynomially injective if it is polynomial and injective in the category of polynomial modules.
Now suppose that has prime characteristic . Let . We write for the set of all column -regular dominant weights. We have the Frobenius morphism , taking an invertible matrix to . If is a rational -module affording representation then write for the vector space viewed as a -module via the representation . We write (resp. , resp. ) for the th infinitesimal subgroup of (resp. , resp. ). The modules , , form a complete set of pairwise non-isomorphic irreducible -modules. Given a rational -module the fixed subspace is a -submodule. A rational -module which is trivial as a is isomorphic to for a uniquely determined (up to isomorphism) rational -module . For a rational -module we have the exact sequence, [[13], II 9.23],
[TABLE]
This will be particularly useful for us in conjunction with the isomorphism
[TABLE]
for and (which follows from the fact that induces an isomorphism ). We shall refer to the first property as the -term exact sequence and use the second property without further reference.
Finally, for , , we have (see [[13], II Proposition 3.16]). We shall also use this without further reference.
3 Some injective indecomposable modules
**Remark **We place the emphasis throughout on injective modules rather than the projective modules as used in [14]. We note that there are certain problems with the argument of Miemietz and Oppermann, [14]. In particular [14], Lemma 4.5 is not correct. (For a counterexample take , , and . The assertion is then that is a certain tilting module but it has highest weight and the tilting module of this highest weight must have the corresponding induced module, i.e the symmetric power , as a section. This is impossible since has dimension and has dimension .) Furthermore, [14], Lemma 4.5 is used in the arguments to justify [14], Lemmas 4.6 and 4.8. (However, we know that the statements are true because of Lemma 3.4 below which we prove using our infinitesimal methods.) Lemmas 4.6 and 4.8 of [14] are used to reach the main goal of the paper [14], namely Theorem 4.13.
For the most part our arguments proceed via the relationship between the representation theory of and and are very different from those in [14]. In order to give a clear line of development we give complete proofs but note that the proof of Lemma 3.4 below has features in common with the proof of [14], Lemma 4.6.
For a -module, resp. -module, we write (resp. ) for the socle of as a -module (resp. -module).
Lemma 3.1**.**
Let be rational -modules. If has -socle , with , then has -socle .
Proof.
Let . Using [[13], II 3.16] we have
[TABLE]
However, as a -module is a direct sum of copies of and so its -socle is a direct sum of copies of . Hence a non-zero term in the above can only occur in case . Moreover, we have
[TABLE]
The result follows.
∎
Lemma 3.2**.**
Let and .
(i) The -module has -socle .
(ii) If is a finite dimensional polynomial -module such that (as -modules) and is injective. Then .
(iii) For the -module has simple socle and if is injective then
[TABLE]
Proof.
(i) Using [[13], II 3.16, (1)], we have an isomorphism of -modules,
[TABLE]
Since has simple -socle we must have
[TABLE]
for some -module with simple socle . Thus embeds in and so embeds in . On the other hand has simple socle by the above Lemma. So embeds in and hence in . So we must have and we are done.
(ii) Note that is indecomposable so is polynomial of some degree , say. To prove that is polynomially injective we show that for all . We write with , . Then . Since is injective the -term exact sequence gives
[TABLE]
The -socle of is so if then and then implies that so that and the above becomes .
(iii) The first assertion follows immediately from Lemma 3.1
We now assume is injective. To conclude we need to show that is polynomially injective and we do this by proving that
for all . As usual we write , with , . We use the term exact sequence, as in the proof of (ii), and we deduce that
[TABLE]
But if then we have and
[TABLE]
since and is polynomially injective.
∎
Remark 3.3**.**
It would be interesting to know precisely for which the module is injective as a -module. We will encounter this property in various special cases in what follows.
Lemma 3.4**.**
Let . Let , for , and . We put and .
(i) The module has socle .
(ii) We have
[TABLE]
(iii) We have
[TABLE]
Proof.
(i) This follows from Lemma 3.1 with , and induction.
(ii) The result is trivial for . Now assume . Let . Let be a composition series of , with , with and , for . Then we have a filtration of . Hence we have
[TABLE]
But now , as a -module, is a direct sum of copies of and , as a -module, is a direct sum of copies of . Hence implies that , and since in this case we must have and
[TABLE]
where . Hence we have
[TABLE]
i.e.,
[TABLE]
and the result follows by induction.
(iii) We identify
[TABLE]
with a subalgebra of via the natural embedding. It suffices to show that the dimension of the second algebra is bounded above by the dimension of the first. By (i), has simple -socle and so, by left exactness of we have that , for . Hence the dimension is bounded above by the composition multiplicity . The result now follows by part (ii).
∎
We introduce some additional notation for use in the proof of the next result. For there exists a unique (up to isomorphism) irreducible -module with highest weight . We write for the injective hull of as a -module. For we have the induced modules and . The character of these modules is given by
[TABLE]
The module admits a filtration with sections , , and also a filtration with sections , .
For we have . We write for the restriction of to . Then is the injective hull of , as a -module, for . For we write (resp. ) for the restriction of (resp. ) to . The module admits a filtration with sections , , and also a filtration with sections , .
For a full discussion of these properties see, [[13], II Chapter 9].
Remark 3.5**.**
Let and with . It is easy to check that .
Lemma 3.6 (iv) and Lemma 4.1(ii) follow from (a very special case of) a result of Andersen, Jantzen and Soergel, [[1], 19.4 Theorem a)]. We provide an independent proof in our case using methods in keeping with the spirit of this paper.
Lemma 3.6**.**
Suppose and . Then:
(i) ;
(ii) ;
(iii) ; and
(iv) is a symmetric algebra of dimension .
Proof.
Let denote an indecomposable summand of which has the highest weight . We shall use the arguments of [[8], p. 236-7], proved for a semisimple group, but the conversion to the context here is routine. (See also [[13],II, 11.10].) It is shown that , that , where is the orbit sum , and . It now follows from Lemma 3.2(ii) that . We have now proved (i) and (ii).
By character considerations, has a filtration with sections , (each occurring once) and a filtration with sections , (each occurring once). Hence has a filtration with sections , (each occurring once) and a filtration with sections , (each occurring once). Moreover, by [[13], II, 9.9 Remark 1], for , we have
[TABLE]
It follows that, for we have
[TABLE]
Hence we have
[TABLE]
However, we may make a similar analysis of . By Brauer’s formula, [[13], II, 5.8 Lemma b)], we have
[TABLE]
Since is a tilting module there is a -module filtration with sections , (each occurring once) and a filtration with sections , (each occurring once). Moreover, by [[13], II, 4.13 Proposition], for , we have
[TABLE]
Hence we have
[TABLE]
Since we must have . This proves (iii) and part of (iv). Finally, putting and writing for the first infinitesimal subgroup we have as the scalar matrices act on via scalar multiplication. Moreover the restriction of an injective -module to is an injective module. The category of -modules is equivalent to the category of modules for the algebra of distribution algebra of , i.e., the restricted enveloping algebra of the Lie algebra . However, since is a symmetric algebra, [11], a finite dimensional injective module for a symmetric algebra is also projective and has symmetric endomorphism algebra so we are done.
∎
Remark 3.7**.**
The above Lemma is consistent with the conjecture, [[5], (2.2)] that a projective indecomposable -module (for a semisimple, simply connected group) is the restriction to of a certain tilting module. It is also consistent with the conjecture [[9],] Conjecture 5.1] which attempts to describe the -modules both injective and projective in the polynomial category.
Proposition 3.8**.**
Let with , for , and let . Then the -module
[TABLE]
is isomorphic to .
Proof.
This follows by induction, using Lemma 3.6(ii) and Lemma 3.2(iii). ∎
**Notation **Let and write with all . We define the -adic breadth to be maximum value of , .
Corollary 3.9**.**
Let and with all , then
[TABLE]
is the injective hull of in the category of polynomial -modules.
Moreover, we have
[TABLE]
Proof.
This follows immediately from Lemma 3.4(iii) and Proposition 3.8. ∎
4 Some Endomorphism Algebras
For we set . In the proof following we write for , write for the tilting module of highest weight , etc. to emphasize dependence on .
Lemma 4.1**.**
(i) For and there is a surjective algebra homomorphism .
(ii) For we have .
Proof.
(i) We denote by the subset of the set of simple roots . Let be the corresponding Levi subgroup. Thus , where is the subgroup of consisting of all invertible matrices with entry [math] for or and and -entry , and is the subgroup consisting of all invertible scalar matrices with entry for . Then has maximal torus and system of positive roots . The set of dominant weights consists of the elements such that . For we write for the simple -module of highest weight , write for the corresponding induced module and for the corresponding tilting module.
We have the truncation functor as in [5]. This functor induces an epimorphism , [[5], (1.5) Proposition]. However, identifying with in the obvious way we have and so , where is a one dimensional -module (of weight given by ). Hence we have and we are done.
(ii) We argue by induction on . For this is clear. Now assume and the result holds for . We have by Lemma 3.6(i). We put and
[TABLE]
By (i) we have an epimorphism . Now
. But
[TABLE]
and since is one dimensional, we have
[TABLE]
By the inductive assumption we therefore have an epimorphism
. Moreover, by Lemma 3.6(iv), the dimension of is so the kernel of this epimorphism, say , is one dimensional. We write . Let be an element mapping to . Then the elements are linearly independent and for some scalar . If then is isomorphic to the truncated polynomial algebra . So we may assume . Since is in the socle of we have . Now and are independent elements of the socle of , but this is impossible since is a local algebra and symmetric, by Lemma 3.6(iv).
∎
We shall need to discuss at some length modules whose endomorphism algebra is a truncated polynomial algebra. To facilitate the discussion in this paper we now introduce appropriate terminology.
**Definition **We say a finite dimensional polynomial -module is admissible of index if its endomorphism algebra is isomorphic to the truncated polynomial algebra . We note that if polynomial -modules are admissible with indices and then is admissible of index .
We assume from now on that .
In the proof of the next result we shall use the following general result, [[13], II proof of proposition 11.6] : if is a group-scheme over a field and is a normal subgroup scheme, if is an -module which is injective as an -module and if an -module which is injective as an -module then is an injective -module.
Proposition 4.2**.**
Let .
(i) For the module is injective as a -module.
(ii) For we have
[TABLE]
Proof.
(i) We write with , and put and . Then is injective as a -module by Lemma 3.6(ii). Thus we have . Now by induction we may assume that is injective as a -module. The Frobenius morphism induces an isomorphism and it follows that is injective as a -module. Hence by the above general remark is injective as a -module.
(ii) Since the two modules in question have the same -socle it is enough to note that is polynomially injective. This is clear from part (i) and Lemma 3.2(ii).
∎
We now show that the representation dimension of grows with . We choose such that . Let be a positive integer. Suppose that is a positive integer which is large enough so that
[TABLE]
We write
[TABLE]
with for and . Then
[TABLE]
For we define by
[TABLE]
and put .
Then
[TABLE]
belongs to .
By Proposition 3.8, Proposition 4.2(ii) and Lemma 4.1(ii) we have that each is admissible of positive index. Note that . (For a general quasi-hereditary algebra over with poset and a maximal element of the corresponding costandard module is injective, see [[7], Definition A 2.1].) Thus has trivial endomorphism algebra so is admissible of index [math]. By Lemma 3.4(iii), to show that is admissible of index at least , it is enough to prove that
[TABLE]
We prove first that
[TABLE]
By Proposition 4.2 (ii), for each with we have that . Hence, extracting all such determinant factors, for a suitable non-negative integer we have
[TABLE]
where if and if . For each we have that , hence by Corollary 3.9 we get
[TABLE]
Moreover by Proposition 4.2 (i), the last module is injective as -module. Therefore is injective as -module and since it has -socle and we get immediately by Lemma 3.2 (ii) that
[TABLE]
This completes the proof of . Finally by Lemma 3.2 (iii) we get that
[TABLE]
We summarise our findings.
Theorem 4.3**.**
Choose such that . Let be a positive integer. Suppose that is a positive integer which is large enough so that
[TABLE]
Then there exists such that is isomorphic to a truncated polynomial algebra with .
Now if is a finite dimensional injective -module then the contravariant dual of is projective and is isomorphic to the opposite algebra of . However, we have, by [14], Corollary 3.3, (which follows from work of Bergh, Oppermann and Rouquier) that if is a finite dimensional algebra over a field with a projective module whose endomorphism algebra has the form with then has representation dimension at least . Hence we have the following.
Corollary 4.4**.**
Choose such that . Let be a positive integer. If is a positive integer large enough so that
[TABLE]
then has representation dimension at least .
5 Some remarks on the quantum case
Generalities
Now let be a field and let be a non-zero element of . We consider the corresponding quantum general linear group , as in [7]. Further details of the framework and proofs or precise references for the results described below may be found in [7]. We have the bialgebra . As a -algebra this is defined by generators , , subject to certain quadratic relations (see e.g., [7], 0.22). Comultiplication and the augmentation map are given by and , for . The algebra has a natural grading such that each has degree . Each component is a finite dimensional subcoalgebra and the dual algebra is the Schur algebra .
The quantum determinant is a group-like element and has an Ore localisation . The bialgebra structure of extends uniquely to a bialgebra structure on the localisation and this localised bialgebra is in fact a Hopf algebra. The quantum general linear group is the quantum group whose coordinate algebra .
We write for the (quantum) subgroup of whose defining ideal is generated by all with . We write for the subgroup of whose defining ideal is generated by all with . By a left (resp. right) module for a quantum group with coordinate algebra we mean a right (resp. left) -comodule. By a -module we mean a left -module. For each there is a one dimensional -module with structure map taking to , where is the defining ideal of . Moreover the modules , , form a complete set of pairwise non-isomorphic simple -modules and the restrictions of these modules to form a complete set of pairwise non-isomorphic simple -modules. All -modules are completely reducible. An element is a weight of a -module if it has a submodule isomorphic to .
Given a subgroup of a quantum group over and an -module we have the induced -module . For the induced module is non-zero if and only if . We set , for . The module has a unique irreducible submodule which we denote . The modules , , form a complete set of pairwise non-isomorphic irreducible -modules.
Suppose is a quantum group over and is a -module with structure map and basis , . The corresponding coefficient elements , are defined by the equations
[TABLE]
The coefficient space of is the -span of all (it is independent of the choice of basis). A -module is polynomial (resp. polynomial of degree ) if (resp. ). A -module which is polynomial of degree may be regarded as an -comodule and hence as a module for the dual algebra . In this way one has equivalences of categories between the category of polynomial -modules of degree , the category of right -comodules and the category of left -modules.
Taking one recovers the classical case of the general linear group scheme and its representation theory. We shall write for , write for and write for in this case. Further, the coordinate function will be denoted in this case.
If is not a root of unity or has characteristic [math] and , then all -modules are completely reducible. We shall assume from now on that is a primitive th root of unity, with . There is a Hopf algebra homomorphism taking to , for . The Frobenius morphism is the quantum group morphism whose comorphism is . By abuse of notation we also write for the restriction of the Frobenius morphism. The infinitesimal group scheme is the subgroup scheme of whose defining ideal is generated by the elements , . For a -module , with structure map we write for the vector space now regarded as a -module via the structure map , where denotes the identity map on . We write for the set of column -regular weights, i.e., the set of such that . The modules , , form a complete set of pairwise non-isomorphic irreducible -modules, see [7], Section 3.2. Moreover, these modules are Schurian, in the sense that , for .
To save on notation we shall try to suppress and where possible. We shall write for , write for , write for and write for . We shall write for the thickening of , i.e., quantum subgroup of whose defining ideal is generated by the elements , . We shall write for and for , .
We shall write for and for . Moreover, for we shall write for and for the socle of . An element of may be written uniquely in the form , with and . Moreover one has (Steinberg’s tensor product theorem, see [7], Section 3.2).
For we shall write for the induced module . The modules have properties analogous to the corresponding modules for reductive algebraic groups, see [13], II. Chapter 3. These modules are considered by Cox in [3], Section 3. There is no explicit statement in [3], Section 3 that these modules are finite dimensional. In a fuller treatment one would include the precise result, namely that the dimension is but we content ourselves here with finite dimensionality. So we show that is finite dimensional for a finite dimensional -module. It suffices to show that the -socle is finite dimensional since is a finite quantum group. Hence it suffices to show that is finite dimensional, for , . Now we have
[TABLE]
using the tensor identity. Hence it suffices to prove that
is finite dimensional for finite dimensional. By left exactness if suffices to prove that is finite dimensional, for . Also, for , with , we have , by [3], Theorem 3.4 (i). Hence we may assume .
Now by [6], Proposition 1.5, (i) (or the original source [16], Section 2.10), , for some -module . We may assume to be non-zero. Let be a non-zero finite dimensional submodule of . Then by Frobenius reciprocity we have
[TABLE]
This is non-zero, (since the left hand side contains the inclusion of in ). Thus we have and so . Note that Frobenius reciprocity gives an embedding of the trivial module in . Let be a copy of in . Then we have . Hence every homomorphism from into goes into , in particular the inclusion has image in . Hence , and so the socle of is and we are done.
The characteristic [math] case
We now consider the case in which has characteristic [math].
Lemma 5.1**.**
If has characteristic [math] then the set of decomposition numbers , , is bounded above.
Proof.
Let be greater or equal to the dimension of , for all . For with , we have , by [3], Theorem 3.4, (i) so that is an upper bound for all , .
Let . We have
[TABLE]
by the transitivity of induction. Let be a -composition series. Thus . By left exactness of induction we have
[TABLE]
A factor has the form for some , . Thus we have
[TABLE]
by [3], Lemma 4.6. Now for and is otherwise [math]. Moreover , for , since -modules are completely reducible. Thus by Steinberg’s tensor product theorem, is either irreducible or [math]. Hence we have . This completes the proof of the Lemma.
∎
Proposition 5.2**.**
For fixed , a field of characteristic [math] and the representation dimension of the Schur algebra (as varies) is bounded above.
Proof.
If and are finite dimensional algebras then the representation dimension of is the maximum of the representation dimensions of and . Hence it suffices to prove that there exists a constant such that the representation dimension of every block of is less than . So let be a block ideal of . Let be the subset of such that the simple modules belonging to the block are the modules , . Let be the projective cover of . Then the block is Morita equivalent to its basic algebra . Representation dimension is a Morita invariant so that it suffices to prove that there exists a constant such that the representation dimension of is less than , for all blocks . But now, from Iyama’s result, [12], 1.2 Corollary, the representation dimension of a finite dimensional -algebra is finite and bounded by a function of the dimension of as a -vector space. Thus it suffices to prove that there exists a uniform bound on the dimension of as ranges over all blocks of (as varies).
Now as a -vector space is isomorphic to . Hence we have , where is the maximal dimension of the spaces , as vary over . Moreover by a result of Cox, [3], Theorem 5.3, we have . Hence it suffices to prove that there is a uniform bound on , as vary over . However, for the algebra is a quasi-heredity algebra with standard modules , (and the dominance order on ). The costandard modules are the induced modules , , and for each the modules and have the same composition factors, counting multiplicities, see e.g., [6], Section 4. Hence we have
[TABLE]
where is an upper bound for all decomposition numbers , for (and the existence of such an is guaranteed by the Lemma above). This completes the proof.
∎
The positive characteristic case
Now suppose that has characteristic . We claim that the main development of Sections 1-4 above goes through in this case.
In addition to the generalities on quantum general linear groups discussed in Section 5.1, we shall need the -analogues of various results described above in the classical case. Our reference for the background results is [7]. Let . In this section we shall write for the (quantum) subgroup scheme of with defining ideal generated by , . In this section is the set of all expressible in the form with an column regular weight and a column regular weight. The modules , , form a complete set of pairwise non-isomorphic -modules (this follows from [3], Lemma 3.1 and Steinberg’s tensor product theorem). The arguments of Section 3 easily adapt to the present context, in particular one may prove a suitable version of Lemma 3.2 (we leave the details to the interested reader).
We pick such that . Let be a positive integer. If we write , with . Let be a positive integer. Suppose that is a positive integer which is large enough so that . We write
[TABLE]
with , for and . We define
[TABLE]
and .
Then belongs to . By the arguments above (using suitable references to [7]), one has that is a tensor product of at least copies of . Hence we obtain the following result.
Theorem 5.3**.**
Suppose is a field of characteristic and is a primitive th root of unity, with . Choose such that . Let be a positive integer. If is a positive integer large enough so that
[TABLE]
then has representation dimension at least .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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