Permutation modules for cellularly stratified algebras
Inga Paul
TL;DR
This paper generalizes the concept of permutation modules from symmetric and Brauer algebras to a broader class called cellularly stratified algebras, including partition algebras, under certain conditions.
Contribution
It introduces a new construction of permutation modules for cellularly stratified algebras, extending previous definitions to partition algebras in suitable characteristics.
Findings
Permutation modules are defined for cellularly stratified algebras.
Partition algebras satisfy the conditions for these modules when the field characteristic is large.
The new framework broadens the applicability of permutation modules in algebra representation theory.
Abstract
Permutation modules play an important role in the representation theory of the symmetric group. Hartmann and Paget defined permutation modules for non-degenerate Brauer algebras. We generalise their construction to a wider class of algebras, namely cellularly stratified algebras, satisfying certain conditions. Partition algebras are shown to satisfy these conditions, provided the characteristic of the underlying field is large enough. Thus we obtain a definition of permutation modules for partition algebras.
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Permutation modules for cellularly stratified algebras
Inga Paul
Institut für Algebra und Zahlentheorie, Universität Stuttgart
Abstract.
Permutation modules play an important role in the representation theory of the symmetric group. Hartmann and Paget defined permutation modules for non-degenerate Brauer algebras. We generalise their construction to a wider class of algebras, namely cellularly stratified algebras, satisfying certain conditions. Partition algebras are shown to satisfy these conditions, provided the characteristic of the underlying field is large enough. Thus we obtain a definition of permutation modules for partition algebras.
Keywords. cellular algebras, permutation modules, Young modules, partition algebras
1. Introduction
The Specht modules are cornerstones of the representation theory of symmetric groups . In characteristic zero, they form a complete set of simple modules ([Jam76, Theorem 3]). In arbitrary characteristic , the simple modules occur as top quotients of Specht modules, in case is a -regular partition111For -singular partitions , is zero. of ([Jam76, Theorem 2]). In the more general case of cellular algebras, introduced by Graham and Lehrer [GL96] in 1996, the cell modules take the role of Specht modules or their duals .
Another cornerstone in the representation theory of symmetric groups are the permutation modules . By James’ Submodule Theorem ([Jam76, Theorem 1]), has a unique direct summand , called Young module, containing as a submodule. Since Young modules are self-dual (cf. [Erd93, 2.2.1 (b)]), can also be characterised as the only direct summand of with quotient . Young modules for different partitions are non-isomorphic ([Jam83, Theorem 3.1 (iii)]). All direct summands of are Young modules , with and appears exactly once ([Jam83, Theorem 3.1 (i)]).
Cellularly stratified algebras, introduced by Hartmann, Henke, König and Paget ([HHKP10]) in 2010, are cellular algebras with additional structure. The aim of this article is to generalise the well-known results about permutation modules for symmetric groups to cellularly stratified algebras containing group algebras of symmetric groups, or their Hecke algebras, as subalgebras. Young modules for cellularly stratified algebras have already been used in [HHKP10]. They were defined abstractly via iterated universal extensions. While this definition is useful for theoretical considerations, the construction of iterated universal extensions might be hard in examples. Extending the construction of Young modules for Brauer algebras of Hartmann and Paget ([HP06]), we present an explicit construction of Young modules (Theorem 1), which coincides, under additional assumptions stated in Section Assumptions, with the abstract definition in [HHKP10] (Corollary 15). This provides new proofs for some results of [HHKP10], e.g. a method of finding all indecomposable (relative) projective modules ([HHKP10, Proposition 12.3]) and Schur-Weyl duality ([HHKP10, Theorem 13.1]). The fact that two Young modules with different indices are non-isomorphic follows from the construction (Corollary 11).
The structural main result of this article is the decomposition of permutation modules into Young modules (Theorem 4). In order to decompose permutation modules for symmetric groups, James used Schur algebras via Schur-Weyl duality and PIMs. There is a Schur-Weyl duality between cellularly stratified algebras and certain quasi-hereditary algebras, which can be regarded as Schur algebras associated to the cellularly stratified algebras, by [HHKP10, Theorem 13.1].
Our homological main result is to show that the Young modules admit filtrations by cell modules (Theorem 2) and are relative projective in the category of modules admitting cell filtrations (Theorem 3). These statements hold provided the cellularly stratified algebra satisfies the additional assumptions stated in Section Assumptions. This generalises a result from Hemmer and Nakano [HN04, Proposition 4.1.1] for Hecke algebras and enables us to prove the analogue of James’ theorem on the decomposition of permutation modules.
This article was inspired by the results of Hartmann and Paget [HP06] for Brauer algebras. We apply the theory developed here to Brauer algebras (Section 5.1) and recover their results (Theorem 5), thus providing new proofs.
Further applications to partition algebras (Section 5.2) show that, provided the characteristic of the field is large enough, we can construct permutation modules for partition algebras with the desired properties (Theorem 6). In order to have the homological Hemmer-Nakano-type results, we need filtrations of restrictions of cell modules to symmetric groups ([Pau16, Theorem 1]) and filtrations of restrictions of permutation modules to symmetric groups. In Proposition 21 we show that the restriction of a permutation module to a group algebra of a symmetric group is isomorphic to a direct sum of permutation modules over this symmetric group. You can find an example (I) and a GAP algorithm (II) to compute the occurring permutation modules in the Appendix.
The approach fails for BMW algebras, the third main example for cellularly stratified algebras in [HHKP10], since the appearing Hecke algebras are not subalgebras of BMW algebras. However, this is satisfied for -Brauer algebras, another deformation of Brauer algebras, and there is hope that the theory applies in this case.
2. Preliminaries
Let be an algebraically closed field, a natural number and an associative -algebra. We denote the symmetric group on letters by ; its Iwahori-Hecke algebra is denoted by , for some unit . Let be the smallest integer such that . If , then . If is an th root of unity, then .
Definition 1** ([HHKP10], Definition 2.1).**
An algebra is called cellularly stratified if the following holds.
- (1)
For each there is a cellular algebra and a vector space such that as a vector space, respecting within each layer the multiplication of , i.e. is an iterated inflation of the cellular algebras along the vector spaces as defined in [KX99]. 2. (2)
For all there are elements such that is an idempotent and for all .
The tuple is called stratification data of .
It follows from the first part of the definition that is cellular with a chain of two-sided ideals
[TABLE]
such that as a non-unital algebra ([KX99, Proposition 3.1 and § 3.2]) which we call the th layer of , and ([HHKP10, Lemma 2.2]). The product of and lies in , where by [KX99, § 3.2].
Remark*.*
If is cellularly stratified and the input algebra is isomorphic to a subalgebra of , then can be regarded as an element , where and are the vectors from the definition of . In this case, we have
[TABLE]
since by a remark on page 5 of [HHKP10], where is the bilinear form defining the multiplication in the inflation, cf. [KX99, § 3.1].
Proposition 2**.**
Let be cellularly stratified such that is isomorphic to a subalgebra of for some . If for all , the algebra is cellularly stratified with stratification data where is a subspace such that , i.e. .
Proof.
Let . Then
[TABLE]
where the inclusion holds up to the isomorphism . Hence we have for some . It is for all . Since , we have for by the remark above, so . Hence . ∎
Our main example of a cellularly stratified algebra will be the partition algebra . It is defined as follows.
Definition 3**.**
Let be an algebraically closed field of arbitrary characteristic. Let and . The partition algebra is the algebra with basis given by all set partitions of . To each set partition, we associate an equivalence class of diagrams consisting of two rows of dots each. Two dots and are connected via a path if and only if they belong to the same part of the set partition. Two diagrams are equivalent, if they correspond to the same set partition.
Example**.**
The set partition corresponds to the diagram
\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}
with path as well as to the diagram
\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}
with path , and the diagrams are equivalent to each other.
We choose to write all diagrams such that the paths are ordered decreasingly with respect to the order , like in the first diagram of the above example. Multiplication is given by concatenation of diagrams. Parts which are not connected to either top or bottom row (called inner circles) are replaced by a factor .
Example**.**
Let
\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet}
and
\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}
in then we have
\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}
\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}
For further details (in a non-diagrammatic setting), see for example [Xi99].
Note that multiplication of diagrams can decrease the number of propagating parts, i.e. parts connecting top and bottom row, but never increase the number of propagating parts.
A diagram consisting of only one row with dots and arbitrary connections is called partial diagram. We have to distinguish certain parts from others; we say they are labelled and write the dots as empty circles instead of dots . When we complete a partial diagram to a full diagram with two rows of dots, the labelled parts become propagating, i.e. they are connected to the other row. We count the parts from left to right, according to the leftmost dot of each part. Let be the vector space with basis all partial diagrams with exactly labelled parts (and possibly further unlabelled parts). For example, \textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ}$$\textstyle{\bullet} is a basis element of , with ; the labelled singleton is the first labelled part, the part is the second. We write to denote the top row of a diagram and for its bottom row. The permutation induced by the propagating parts is denoted by . It is well-defined by the convention to connect labelled top and bottom row parts via their respective leftmost dots.
If , the partition algebra is cellularly stratified by [HHKP10, Proposition 2.6] with stratification data . The idempotents are given by
[TABLE]
For , there is an algebra isomorphism given by attaching dots to the right of both top and bottom row and connecting the new dots to the rightmost dots of top and bottom row respectively of the original diagram.
The partition algebra contains the Brauer algebra and the group algebra of the symmetric group as subalgebras. The Brauer algebra is the subalgebra with basis given by all diagrams where each dot is connected to exactly one other dot. We call such a connection (horizontal) arc if it connects two dots within the same row. A permutation corresponds to the diagram connecting the th dot of the top row to the th dot of the bottom row.
2.1. Setup
Let be cellularly stratified with stratification data where the are isomorphic to group algebras of symmetric groups or their Iwahori-Hecke algebras, such that for each we have an embedding of algebras. This is satisfied for Brauer algebras and partition algebras, but not for BMW-algebras, the third main example of cellularly stratified algebras in [HHKP10]. However, it is satisfied for another deformation of Brauer algebras: the -Brauer algebras defined by Wenzl in [Wen12]. We choose as cell modules for the cellular algebras the dual Specht modules .
We need two types of induction and two types of restriction functors, namely
[TABLE]
where denotes the two-sided ideal and is a short notation for .
Remark*.*
has a right -module structure because we assumed to be isomorphic to a subalgebra of . Any -module has an -module structure via the quotient map , cf. [HHKP10, Lemma 2.3].
Let . We call the left-ideal the layer of . The functor sends a -module to an -module living in the layer, i.e. . This is explained in the beginning of Subsection 2.2. We call this functor layer induction.
The induction functor sends a -module to an -module with non-zero action of , i.e. lives in all layers with .
While removes the lower layers (with ) of the -module , keeps all layers of the module.
2.2. Properties of the Functors
For each -module , we have , where acts on both and via ([HHKP10, Lemma 2.3]). Thus, the layer induction corresponds to the functor , defined in [HHKP10]. Hence, we can apply [HHKP10, Lemma 3.4] to get an isomorphism of -modules. We will make extensive use of the isomorphisms
[TABLE]
without special mention.
Proposition 4** ([HHKP10], Propositions 4.1 - 4.3; Corollary 7.4; Propositions 8.1 and 8.2).**
The functor has the following properties.
- (1)
It is exact. 2. (2)
The set is a complete set of cell modules for . 3. (3)
* for all .* 4. (4)
* for all and .* 5. (5)
* for all and .*
If then
- (6)
* for all .* 2. (7)
* for all and .*
The induction is not exact in general and does not send cell modules to cell modules. However, we will give sufficient conditions for to send cell filtered modules to cell filtered modules in Section 3. Theorem 3 will tell us that, under additional conditions, sends relative projective modules to relative projective modules, cf. Definition 7.
The following properties of the functors are straightforward calculations. The layer restriction is right-exact, but in general not exact. It is left adjoint to and left inverse to both and . The restriction is exact, since is projective as right -module. It is left adjoint to and right adjoint to , i.e. we have a triple of adjoint functors. Furthermore, is left inverse to , but in general not to ; the layers added by are not removed by .
For example, if is the Brauer algebra with and , and is the trivial -module , then , which consists of all linear combinations of Brauer diagrams with exactly one horizontal arc per row. The left -module contains which has a basis
[TABLE]
where the brackets denote residue classes containing all three bottom row configurations. In particular, is non-zero and not isomorphic to .
Proposition 5**.**
If is a cell module of , then is a cell module of or zero.
Proof.
Let be a cell module of . By Proposition 4, part (2), we have for some , where is a dual Specht module in . This implies , where . If , then and if , then . So, in both cases we have . For , we have . Thus, the layer restriction of a cell module from the same layer is a cell module, while cell modules from other layers vanish under restriction. ∎
2.3. Further Definitions and Notation
Let , where is the index of the symmetric group related to and means that is a partition of . We define an order on by setting
[TABLE]
Let and let be the corresponding permutation module in .
Definition 6**.**
We call the -module permutation module for .
Let denote the set of cell modules. The category of -modules with a cell filtration, i.e.smodules admitting a chain of submodules such that the subquotients are isomorphic to cell modules, is denoted by . The category of -modules admitting a filtration by dual Specht modules is denoted by .
Definition 7** ([HHKP10], Definition 11.2).**
Let . We say that is relative projective in , if
for all .
is the relative projective cover of , if is minimal with respect to the property that there is an epimorphism with .
3. Young Modules
In this section, we define Young modules as direct summands of permutation modules, following the definitions given for Brauer algebras by Hartmann and Paget, [HP06]. This allows us to extend the results of James for group algebras of symmetric groups to cellularly stratified algebras whose input algebras are isomorphic to group algebras of symmetric groups or their Hecke algebras.
Theorem 1**.**
Let be a cellularly stratified algebra with input algebras isomorphic to group algebras of symmetric groups or their Hecke algebras. Assume that as -bimodules. Then has a unique direct summand with quotient isomorphic to .
Proof.
It is well-known that the -permutation module decomposes into a direct sum of indecomposable Young modules with multiplicities , where and implies ([Jam83, Theorem 3.1]). Therefore, we have . Decompose further into a direct sum of indecomposables for .
Claim 1**.**
has a direct summand with quotient isomorphic to .
Let be the projection onto and the inclusion of . The functor is exact, so applying it to the composition gives maps
[TABLE]
By assumption, we have a decomposition of right -modules. Thus the homomorphism is given by a matrix, where the top left entry is an endomorphism . This gives a commutative diagram
[TABLE]
Let . Let for some . By lower terms we mean terms of the form with and .
The commutativity of the above diagram says that, up to isomorphism, we have , so we have .
The identity on is , so
[TABLE]
for any . Since there are no lower terms on the left hand side, they vanish on the right hand side and we have . Hence , i.e. is the identity on .
Let . Then , so for any we have . Therefore, . is finite dimensional and indecomposable, so is local. Thus for all , either or is a unit. To show that at least one is a unit, assume that are non-units. Then is a unit. We now assume without loss of generality that is a unit, in particular surjective.
Let
[TABLE]
and its restriction to . Then
[TABLE]
since for and , . Surjectivity of implies that the -homomorphism is surjective, so is a quotient of .
Claim 2**.**
is the only summand of with quotient isomorphic to .
Suppose there is another summand of such that there is an epimorphism with and for all . By tensor-hom adjunction, is an element in , so is given by
[TABLE]
for some . For and , we have
[TABLE]
The surjectivity of provides the existence of a preimage of with and for all . Since decomposes into as -bimodule, we can write any element as with and . Thus for some . So sends to
[TABLE]
On the other hand,
[TABLE]
so , hence . But and
[TABLE]
since and is a unit, in particular injective. So , which contradicts the definition of .
Claim 3**.**
There is no summand of with quotient for .
Assume there is a direct summand of with such that is a quotient of . An arbitrary homomorphism is given by for some by the adjunction . is surjective only if is surjective222Assume there is such that for all . Let be an arbitrary element of and suppose that . Then for all and .
The rest of the proof can be copied from [HP06] in case . We give here a similar proof for Iwahori-Hecke algebras , inspired by the one for group algebras of symmetric groups, using notation from [DJ86].
Suppose there is an epimorphism , which we extend to an epimorphism such that is zero on all summands other than , i.e. is the projection from onto the direct summand , followed by the map . Recall (e.g. from [DJ86]) that is generated by elements , and , where . For , where is the length function on symmetric group elements and is the conjugate of the partition , we have that implies by [DJ86, Lemma 4.1]. So for , we have . Then . But contains the generator of , in particular .
This concludes the proof of Theorem 1. ∎
Definition 8**.**
We denote the unique summand of with quotient constructed above by , in analogy to [HP06], and call it Young module for with respect to .
We now collect conditions for a Young module to appear as a summand of . They generalise the conditions from [HP06, Lemmas 17 and 18] for . The fact that these are the only direct summands of permutation modules is our main result (Theorem 4) and will be proven using results from the next Section.
Lemma 9**.**
If with , then does not appear as a summand of .
Proof.
is left adjoint to , so
[TABLE]
For , , so . Thus, there cannot be a non-zero map
[TABLE]
since it would extend to a non-zero map . ∎
Lemma 10**.**
If , then occurs as a direct summand of if and only if is a direct summand of . This can only occur if .
Proof.
If is a direct summand of , then , as a direct summand of , is a direct summand of .
If is a direct summand of and , then is a summand of for some .
It follows from Theorem 1, Claim 3, that , so is a direct summand of . ∎
Corollary 11**.**
If , then .
Proof.
Let . Then , see for example [Mar93, Section 7.6], so since otherwise would be isomorphic to . Assume that . Then is a direct summand of and by Lemma 10, is a direct summand of . So is a summand of and has a summand with quotient . But is isomorphic to with quotient , so has a direct summand with quotient isomorphic to and . This contradicts Claim 3 from Theorem 1. ∎
4. Properties
Each Young module is a direct summand of the permutation module by definition. In this section, we show that the indecomposable direct summands of permutation modules are exactly the Young modules, as in the symmetric group case. The results extend the results on Brauer algebras stated in [HP06] to our setup.
We give conditions under which the permutation modules for our cellularly stratified algebra admit a cell filtration in Subsection 4.1. In Subsection 4.2, we show that permutation modules are relative projective in the subcategory of cell filtered -modules, provided a further condition is satisfied. Then the Young module is the relative projective cover of the cell module (Theorem 3). As a corollary of this, we recover a result about Schur-Weyl duality from [HHKP10] in Subsection 4.3. Finally, we can prove Theorem 4, the decomposition of the permutation module into a direct sum of Young modules , in Subsection 4.4.
A crucial point in the study of a category of -filtered -modules is that it is closed under direct summands if the set with ordered index set forms a standard system333cf. [DR92, Section 3] or [HHKP10, Definition 10.1], i.e. for all
is a division ring.
implies .
implies .
The statement follows from [Rin91, Theorem 2].
Lemma 12**.**
Let be as defined in Subsection 2.1. Let if the input algebras are group algebras of symmetric groups and let if the input algebras are isomorphic to Hecke algebras . Then the cell modules of form a standard system with respect to the order defined in Subsection 2.3.
Proof.
Dual Specht modules for symmetric groups form a standard system by [HN04, Proposition 4.2.1] and [Jam78, Corollary 13.17]. Dual Specht modules for Iwahori-Hecke algebras of symmetric groups form a standard system by [HN04, Proposition 4.2.1] and [Mat99, Exercise 4.11]. The statement follows from [HHKP10, Theorem 10.2 (a)]. ∎
Assumptions**.**
We give names to the following assumptions that we make on in order to prove the desired properties for permutation modules and Young modules. Furthermore, we often assume that (or , in case the are Iwahori-Hecke algebras) to be able to use Lemma 12.
Let be as defined in Subsection 2.1 and let .
-
(I)
-
(a)
as -bimodules and 2. (b)
as right -modules. 2. (II)
as right -modules. 3. (III)
Layer-removing restriction to of a permutation module from layer is dual Specht filtered:
[TABLE] 4. (IV)
Classical restriction to of a cell module from layer is dual Specht filtered:
[TABLE]
Remark*.*
Assumption (Ia) is the assumption we made in Theorem 1 in order to define Young modules for .
Assumption (IV) implies that for any , : The functor is exact and sends dual Specht modules to cell modules, so has a cell filtration. is exact, so has a filtration by modules of the form . The statement follows since is extension-closed.
Lemma 13**.**
*Instead of (II), we can assume
(II’)* as vector spaces.*
Proof.
By Proposition 2, the algebra is cellularly stratified with idempotents . Then is free of rank over by [HHKP10, Proposition 3.5] and as left -modules. Hence, , since is free of rank over .
The multiplication map
[TABLE]
is an epimorphism of -bimodules and by (II’), so (II) is satisfied. ∎
4.1. Cell filtrations
Theorem 2**.**
*Assume that satisfies (I),(II) and (III). Then the permutation module has a filtration by cell modules.
If, in addition, or , then the direct summands of have cell filtrations.*
Proof.
is a filtration of (with quotients isomorphic to ), so we have short exact sequences
[TABLE]
of -bimodules for . Application of the exact restriction functor gives exact sequences
[TABLE]
of -bimodules for , which are split exact as sequences of right -modules by assumption (Ib). Hence, we get exact sequences
[TABLE]
of left -modules, which give rise to a filtration
[TABLE]
of with quotients , the layer of . Assumption (II) gives
[TABLE]
By assumption (III), . The functor is exact and sends dual Specht modules to cell modules by Proposition 4, so for all , in particular .
If is different from and , then the cell modules of form a standard system by Lemma 12. In this case, is closed under direct summands by [Rin91, Theorem 2], so all direct summands of , in particular the Young modules , admit cell filtrations. ∎
4.2. Relative projectivity
An important property of the permutation modules is their relative projectivity in the category , as shown by Hemmer and Nakano in [HN04, Proposition 4.1.1], in case . This property is translated to the permutation modules of , in case the conditions (I) to (IV) are satisfied. Furthermore, the Young modules are relative projective covers of the cell modules.
Theorem 3**.**
Assume that satisfies (I) to (IV). Then the permutation module is relative projective in . If, in addition, (or ), then all direct summands of are relative projective in . Furthermore, is the relative projective cover of in the category of cell filtered modules.
Proof.
By Theorem 2, and all its direct summands (provided or ) are in if satisfies conditions (I) to (III). We have to show that for all . Let and let
[TABLE]
be a short exact sequence in .
Apply the exact functor on to get a short exact sequence
[TABLE]
in . Now we apply the left exact functor to get a long exact sequence
[TABLE]
It follows from assumption (IV) and the exactness of that for . Since is relative projective in , we get , in particular we get a short exact sequence
[TABLE]
which is isomorphic to the short exact sequence
[TABLE]
since is right adjoint to .
Consider
[TABLE]
then exists (such that the diagram commutes) by surjectivity of the map in . This shows that splits and so . In particular, is relative projective in .
Now let be a direct summand of with the projection onto and the inclusion of into . With the same strategy as above, applied to the short exact sequence
[TABLE]
we see that the map is surjective, which provides the existence of a map such that :
[TABLE]
But , so and is right inverse to . Therefore, the sequence splits and , so all direct summands of are relative projective in .
In order to prove that is the relative projective cover of , we have to show that there is an epimorphism
[TABLE]
with and that is minimal with respect to this property. Once we have established the epimorphism, the minimality condition is immediately satisfied since is indecomposable, and then is a relative projective cover of .
The -module has a dual Specht filtration with top quotient , so the kernel of the map lies in . The functor is exact and sends dual Specht modules to cell modules, so the kernel of the epimorphism
[TABLE]
has a cell filtration.
Recall from the proof of Theorem 1 that there is an epimorphism
[TABLE]
Consider the commutative diagram
[TABLE]
with , , and in . The composition
[TABLE]
is zero, so the universal property of the kernel of provides a unique morphism , with kernel , making the diagram
[TABLE]
commutative. The map is given by the universal property of the kernel of and is an isomorphism by the snake lemma. The snake lemma also asserts surjectivity of the map .
Thus, we have a short exact sequence
[TABLE]
with . If we can show that , then since is extension-closed.
Consider the commutative diagram
[TABLE]
We have , so restricts to .
Now, we consider the commutative diagram
[TABLE]
where is the projection from onto its summand . We see that , so restricts to .
[TABLE]
In particular, is a direct summand of . By the proof of Theorem 2, the module has a cell filtration. By the assumption on the characteristic of the field, cell filtrations restrict to direct summands, so and lie in . Since is extension-closed, we get and so is a relative projective cover of . ∎
Corollary 14** ([HHKP10], Corollary 12.4).**
If is a group algebra of a symmetric group for some , , and satisfies (I) to (IV), then is projective if and only if is -restricted.
4.3. Schur-Weyl duality
In [HHKP10], the Young modules of a cellularly stratified algebra are defined as the relative projective covers of the cell modules , in the case where the cell modules of the input algebras form standard systems. Since we assumed to be isomorphic to or for some and , respectively , we are in this situation (Lemma 12). Therefore, we have the following corollary of Theorem 3.
Corollary 15**.**
The Young modules , defined abstractly in [HHKP10], coincide with the explicitly defined Young modules of this article.
In particular, we are in the situation of Theorem 13.1 from [HHKP10]:
Corollary 16**.**
Let be as defined in Subsection 2.1, such that the assumptions(I)* to (IV) are satisfied and let (or ). Then the following holds.*
- (1)
Each has well-defined filtration multiplicities. 2. (2)
The category of cell filtered -modules is equivalent, as exact category, to the category of standard filtered modules over the quasi-hereditary algebra , where
[TABLE]
and 3. (3)
There is a Schur-Weyl duality between and . In particular, we have .
Remark*.*
The multiplicities of the Young modules in are chosen to be minimal such that all Young modules appear at least once and such that the projective Young modules appear as often as they appear in , i.e. such that there is a with .
4.4. Decomposition of permutation modules
Using the results of the previous subsections, we are finally able to prove that permutation modules for decompose into a direct sum of Young modules, just like permutation modules for decompose into direct sums of Young modules.
Theorem 4**.**
Let be as defined in Subsection 2.1, such that the assumptions(I)* to (IV) are satisfied and let (or ). Let . Then there is a decomposition*
[TABLE]
with non-negative integers . Moreover, .
Proof.
By Lemma 12, the set forms a standard system. Corollary 16 says that there is a quasi-hereditary algebra such that the categories of cell filtered -modules and of standard filtered -modules are equivalent, which was first established in [DR92]. To prove this equivalence, Dlab and Ringel show that there is a one-to-one correspondence between the modules in the standard system and the indecomposable relative projective modules in . By Theorem 3, the Young modules are indecomposable relative projective. The one-to-one correspondence shows that these are all indecomposable relative projective -modules, since for each there is exactly one Young module and exactly one cell module, and these are all cell modules, cf. Propostition 4 part (2), Theorem 1 and Corollary 11. The algebra is quasi-hereditary, so the relative projective -modules are exactly the projective -modules, cf. [Rin91, Corollary 2], and they correspond under the equivalence to the relative projective -modules. Hence, the projective -modules are indexed by .
The permutation module is relative projective in , so its image under the equivalence \mathcal{F}(\Theta)\xrightarrow{\,\smash{\raisebox{-2.79857pt}{\sim}}\,}\mathcal{F}(\Delta) is a projective -module . Let be a decomposition of into indecomposable modules. Sending back to through the equivalence, its image must be an indecomposable relative projective module . Thus, for some non-negative integers . by definition of . Lemmas 9 and 10 show that we only have to sum over those Young modules with . ∎
5. Applications
There are three main examples of cellularly stratified algebras in [HHKP10]: Brauer algebras, partition algebras and Birman-Murakami-Wenzl algebras (BMW algebras), a deformation of Brauer algebras. The results for Brauer algebras first appeared in [HP06]. With the theory from this article, we can recover their results, using less combinatorics specific to Brauer algebras but the more structural properties of cellularly stratified algebras, which have been introduced after the work of Hartmann and Paget on Brauer algebras appeared. We recover the results for Brauer algebras in Subsection 5.1, thus providing new proofs. In Subsection 5.2, we show that the results hold for partition algebras under certain additional assumptions. The theory fails for BMW algebras, since we need the cellular algebras to be subalgebras. However, the -Brauer algebras, defined by Wenzl in [Wen12], are another deformation of Brauer algebras which fit into this setting. They are cellularly stratified as shown by Nguyen in his PhD thesis [Ngu13] and contain Hecke algebras as subalgebras. We do not prove that the -Brauer algebras satisfy the assumptions in this article.
5.1. Recovering Results for Brauer Algebras
Let be the Brauer algebra on dots with . If is even, let . Then by [HHKP10, Proposition 2.4], is cellularly stratified with stratification data
[TABLE]
where if is even and if is odd, and is the vector space with basis consisting of partial diagrams with exactly horizontal arcs. The idempotents are defined as e_{l}=\frac{1}{\delta^{\frac{r-l}{2}}}\cdot\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 9.1111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&&\\&&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-9.1111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\overset{1}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 15.00002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 39.66661pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 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0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 99.06935pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 123.73595pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 141.73595pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\overset{r}{\bullet}}}}}}}}}{\hbox{\kern-5.5pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 15.00002pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 42.36798pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 63.06935pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 81.06935pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 99.06935pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 123.73595pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 145.24171pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage} for . For (and odd), we use e_{l}=\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 9.1111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&&&\\&&&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-9.1111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\overset{1}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 15.00002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 39.66661pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 57.66661pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\overset{l}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 81.06935pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 102.40265pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 123.73595pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 148.40254pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 166.40254pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\overset{r}{\bullet}}}}}}}}}{\hbox{\kern-5.5pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 15.00002pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 39.66661pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 60.36798pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 81.06935pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 99.06935pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 127.06924pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 148.40254pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 169.90831pt\raise-16.69443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}.
We want to recover the results from [HP06], so we have to show that the Young modules defined here coincide with those defined in [HP06] as indecomposable submodules of with quotient . The module structure on is defined as follows. Let be a basis element and let . Then
[TABLE]
where is the partial diagram obtained by writing on top of , identifying with and following the new connections in , multiplying by for each closed loop. If the result is not in , set . The permutation is given by the permutation of the free dots of in .
Example**.**
Let b=\begin{minipage}{113.81102pt}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern-5.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 15.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 36.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 57.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}\in B_{k}(4,\delta) and v=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}\ignorespaces}}}}\ignorespaces\in V_{2}. Then bv=\delta^{\#\text{closed loops}}\text{top}\left(\begin{minipage}{113.81102pt}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern-5.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 15.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 36.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 57.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern-5.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 15.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 36.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 57.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces}}}}\ignorespaces\end{minipage}\right)=\delta\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}\ignorespaces}}}}\ignorespaces and .
Proposition 17**.**
For any , there is an isomorphism of -modules.
Proof.
Let and consider the map
[TABLE]
where is the diagram in with and non-crossing propagating lines444Since is in , its bottom row is fixed: free dots followed by horizontal arcs sitting side by side.. Let , with corresponding to under the isomorphism , i.e. . Then , so is surjective. By [HHKP10, Proposition 3.5], . Hence, is bijective. To see that is an isomorphism, we have to check that it is -linear. Let and . Then
[TABLE]
and
[TABLE]
If then . On the other hand, implies that has more than horizontal arcs, so . If has propagating lines, then for some and of the free dots555In this case, a free dot is a dot which does not belong to a horizontal arc. of are bound by horizontal arcs in since the product lies in . The remaining free dots of are end points of propagating lines in . Therefore, the permutation of the propagating lines of is . This shows and is -linear. ∎
Corollary 18**.**
The cell, Young and permutation modules defined here coincide with those defined in [HP06].
It remains to verify that , with if is even, satisfies the assumptions (I) to (IV). Let .
The decompositions and hold for vector spaces. The left (resp. right) action of permutes the dots of the top (resp. bottom) row, but it never changes the amount of horizontal arcs, so assumption (I) is satisfied. Assumption (II) holds by [HK12, Lemma 4.3]. By [HK12, Lemma 4.2], , where . We get the following isomorphisms of -modules
[TABLE]
The last module is equal to a direct sum of -permutation modules by [HK12, Lemma 4.5]. Therefore, and assumption (III) is satisfied. The restriction of a cell module to , with , is dual Specht filtered by [Pag07, Proposition 8], thus satisfies assumption (IV). This gives a new proof for the following theorem.
Theorem 5** ([HP06]).**
Let . The Brauer algebra , with if is even, has permutation modules , which are a direct sum of indecomposable Young modules. The Young modules are the relative projective covers of the cell modules . Every module admitting a cell filtration has well-defined filtration multiplicities.
5.2. New Results for Partition Algebras
Now, let be the partition algebra on dots with . Then is cellularly stratified by [HHKP10, Proposition 2.6]. The stratification data, as well as an isomorphism for , was described in Section 2. We use the following embedding of into . Let be the diagram describing the permutation , i.e. the dot in the top row is connected to the dot in the bottom row, and these are all the connections. Then becomes an element of by attaching dots to the right of the top row and connecting all these new dots to the th dot of the top row. Do the same for the bottom row. This embedding agrees with the isomorphism from Section 2.
Example**.**
Let and let . Then d(\pi)=\begin{minipage}[c]{71.13188pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-5.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 15.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 36.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 57.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage} is clearly an element of . The corresponding element in is \begin{minipage}{85.35826pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&\\&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 78.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 99.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 120.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern-5.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 15.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 36.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}{\hbox{\kern 57.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 78.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 99.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 120.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}.
In particular, for each , the input algebra of the cellularly stratified structure is a subalgebra of .
It remains to show that satisfies conditions (I) to (IV). Fix some between [math] and and remember that denotes the two-sided ideal . Set .
The left (resp. right) action of on a partition diagram permutes the top (resp. bottom) row of , but it never changes the size of a part of . In particular, the number of propagating lines remains invariant under the -action and the decompositions from assumption (I) are indeed decompositions of -(bi)modules.
For , the basis diagrams of have exactly propagating parts and the last dots of the bottom row belong to the same part. Hence, we have an isomorphism of vector spaces , where is the vector space of partial diagrams with exactly labelled parts and is the subspace of where the last dots belong to the same part. This shows assumption (II*′*) is satisfied and thus, by Lemma 13, assumption (II) is satisfied as well.
Assumption (IV) holds by [Pau16, Theorem 1] in case . The condition on the characteristic is sufficient, but potentially too strong, as explained in [Pau16].
We now prove that assumption (III) is satisfied. Fix . When dealing with the size of a part in a partial diagram, we will from now on count the last dots as one. Let . We say that is equivalent to , , if and only if there is a such that , where is defined as follows. Write the diagram below and identify with . Then is the bottom row of this diagram, where a part is labelled if and only if it contains at least one labelled dot. In diagrams, this means that and are equivalent if and only if for each size, the number of labelled parts and the number of unlabelled parts of and coincide. Remember that the last dots count as one.
For , we define to be the diagram in with , and . Let be a diagram with . By definition, there is a such that . Then . Let be the -bimodule generated by .
Lemma 19** ([Pau16, Lemma 1]).**
The -bimodule decomposes into .
Fix a partial diagram and set . Let be the number of labelled parts of size and the number of unlabelled parts of size of , where again the last dots count as one dot. Then and . Without loss of generality, assume that the parts of are ordered as follows. The labelled parts are on the left hand side, the unlabelled parts on the right hand side. The parts are then ordered increasingly from left to right.
Let be the set of dots of belonging to the th labelled part of size and let be the set of dots of belonging to the th unlabelled part of size . Then is the stabilizer subgroup of which stabilizes exactly the labelled parts of . Similarly, the stabilizer subgroup of which stabilizes the unlabelled parts of is . In particular, stabilizes , while permutes the propagating lines of . Note that and , where denotes the wreath product. Define a right-action of on via for and , i.e. acts on via the canonical epimorphism
[TABLE]
Then we can define the tensor product .
Lemma 20** ([Pau16, Lemma 2]).**
There is an isomorphism of -bimodules given by .
We want to understand the summands of for a partition of . Fix double coset representatives of . To each , we attach a composition as follows. Set . Then is in if and only if there is a such that . Since is isomorphic to , it is a Young subgroup of , and is a direct product of wreath products of symmetric groups. Then the intersection is again a product of wreath products. The image of under the canonical epimorphism is a Young subgroup of , which we denote by .
An example for , and a GAP-algorithm to compute them can be found in the appendix.
Proposition 21**.**
The left -module is isomorphic to the direct sum of various permutation modules. In particular, it admits a filtration by dual -Specht modules.
Proof.
We define a map
[TABLE]
as follows. Let and with for some and . Set with non-zero entry only in the summand. Extend this -linearly to get a -homomorphism.
We have to show that this map is well-defined, that is we have to show that whenever two elements and are equivalent in , then their images are equivalent in .
Let with and and let and . Since , we have and . It follows that , so is well-defined.
The inverse is given by
[TABLE]
with for :
[TABLE]
and
[TABLE]
for , and .
It remains to show that is well-defined. Let such that and are equivalent in for some . Then there is a such that . It follows that for some with . By definition of as the image of the canonical projection , we have . So there is a such that . Therefore we have and is well-defined. ∎
Corollary 22**.**
Layer restriction of a permutation module is isomorphic to a direct sum of permutation modules. In particular, layer restriction of a permutation module has a dual Specht filtration.
Proof.
For , we have , so the statement is true. For , we can apply Lemmas 19, 20 and Proposition 21 to get a decomposition
[TABLE]
∎
This shows that assumption (III) is satisfied and we can conclude the following theorem.
Theorem 6**.**
Let and let be an algebraically closed field and let be zero or at least . Then the partition algebra , with , has permutation modules , which are a direct sum of indecomposable Young modules. The Young modules are relative projective covers of the cell modules . Every module admitting a cell filtration has well-defined filtration multiplicities.
Appendix I Example for Proposition 21, calculated by hand
Example**.**
Let v=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ}}}}}}}}{\hbox{\kern 12.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ}}}}}}}}{\hbox{\kern 30.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ}}}}}}}}{\hbox{\kern 48.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 66.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ}}}}}}}}{\hbox{\kern 84.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 102.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ}}}}}}}}{\hbox{\kern 120.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 138.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet}}}}}}}}\ignorespaces}}}}\ignorespaces\in V_{5}^{9}. The summand of is isomorphic to
[TABLE]
This can be verified as follows. We have and the set of double coset representatives is
[TABLE]
The only transpositions in leaving invariant are those with both end points belonging to the same part . In the partial diagram , mark these dots as if they were labelled. The only products of two disjoint transpositions leaving invariant are those where and (or and ) belong to the same part and and (or and , respectively) belong to the same part . Note that here, is possible666In this case, the transpositions and belong to the first group of dots ( or ) as well.. Mark these dots as if they were labelled. Put vertical lines at the end of each part . Translate this back to . We can read off from the given information by the labelling of dots: s of the same size become symmetric groups, s become wreath products, if both end points lie in the same part in and the group generated by otherwise777It does not make a difference for which of the two cases we have, since the projection onto is the same.. We do this for each double coset representative in Table 1.
Appendix II GAP code to compute summands of restriction of permutation modules for partition algebras
For a given summand of , the following GAP code calculates which Young subgroups appear in the decomposition of , given in Proposition 21.
As input, we need G, H and K, as well as the list imgs of images of the generators of under the canonical epimorphism , sending to . We state the code for the example in Appendix I.
INPUT: S2:=SymmetricGroup(2); S3:=SymmetricGroup(3);
S5:=SymmetricGroup(5); S7:=SymmetricGroup(7); # abbreviations
G:= SymmetricGroup(14); H:=DirectProduct(S3,WreathProduct(S2,S2),S2); K:=DirectProduct(S7,S2); # , , .
gens:=GeneratorsOfGroup(H);
imgs:=[(1,2,3),(1,2),(),(),(4,5),()];
to each generator gens[i], set
imgs[i]:=the image of gens[i] # under the canonical epimorphism # .
hom:=GroupHomomorphismByImages(H,S5,gens,imgs);
iso=function(G,H,K)
local L, r, R, Pinu, Snu;
L:=[]; R:=List(DoubleCosets(G,H,K),Representative);
for r in R do
PinuIntersection(H,ConjugateSubgroup(K,r^-1));
SnuImage(hom,Pinu);
Add(L,Snu);
od;
return L;
end;
OUTPUT: list L of all appearing Young subgroups Snu of S5.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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