# Kronecker positivity and 2-modular representation theory

## Abstract

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of which are of 2-height zero.

## Introduction

This paper brings together, for the first time, the two oldest open problems in the representation theory of the symmetric groups and their quiver Hecke algebras. The first problem is to understand the structure of Specht modules and the second is to describe the decomposition of a tensor product of two Specht modules — the *Kronecker problem*.

### Kronecker positivity

The Kronecker problem is not only one of the central open problems in the classical representation theory of the symmetric groups, but it is also one of the definitive open problems in algebraic combinatorics as identified by Richard Stanley in Reference Sta00. The problem of deciding the positivity of Kronecker coefficients arose in recent times also in quantum information theory Reference Kly04Reference CM06Reference CHM07Reference CDW12 and Kronecker coefficients have subsequently been used to study entanglement entropy Reference CSW18.

A new benchmark for the Kronecker positivity problem is a conjecture of Heide, Saxl, Tiep and Zalesskii Reference HSTZ13 that was inspired by their investigation of the square of the Steinberg character for simple groups of Lie type. It says that for any there is always a complex irreducible character of whose square contains all irreducible characters of as constituents. For a triangular number, an explicit candidate was suggested by Saxl in 2012: Let denote the staircase partition. Phrased in terms of modules, Saxl’s conjecture states that all simple modules appear in the tensor square of the simple th -module In other words, we have that .

with for all partitions of This conjecture has been studied by algebraists, probabilists, and complexity theorists .Reference Bes18Reference Ike15Reference LS17Reference PPV16 yet remains to be proved in general. Positivity of the Kronecker coefficient has been verified for hooks and two-row partitions when is sufficiently large in Reference PPV16, and then for arbitrary and a hook in Reference Ike15Reference Bes18 or a double-hook partition (i.e., when the Durfee size is 2) in Reference Bes18, and for any comparable to in dominance order in Reference Ike15.

This paper begins with the observation that the -module is projective over a field of characteristic or equivalently, that the character to the Specht module , is the character associated to a projective indecomposable (via its integral lift to characteristic 0). Therefore, the tensor square of -module is again a projective module, and the square of is the character to a projective module. This allows us to bring to bear the tools of modular and graded representation theory on the study of the Kronecker coefficients. In particular, we deduce that if is a simple Specht module, then all constituents of the projective cover of must also appear in Saxl’s tensor-square. For example, using this property for the trivial simple module of at characteristic 2 gives all characters of odd degree as constituents in the Saxl square; more generally, we will detect all irreducible characters of 2-height 0 as constituents. Our aim is to understand the columns of the 2-modular graded decomposition matrix which are labelled by *simple Specht modules* and to utilise these results towards Saxl’s conjecture.

### Modular representation theory

The classification of simple Specht modules for symmetric groups and their Hecke algebras has been a massive undertaking involving over 30 years of work Reference Jam78Reference JM96Reference JM97Reference JM99Reference Fay04Reference Fay05Reference JLM06Reference Lyl07Reference FL09Reference Fay10Reference FL13, with some conjectural cases for and still to be verified. The pursuit of a description of semisimple and decomposable Specht modules is similarly old Reference Jam78 and yet has proven a much more difficult nut to crack. The decomposable Specht modules labelled by hook partitions were characterised by Murphy and Speyer Reference Mur80Reference Spe14; the graded decomposition numbers of these Specht modules were calculated by Chuang, Miyachi, and Tan Reference CMT04; the first examples of decomposable Specht modules labelled by non-hook partitions were given by Dodge and Fayers Reference DF12; Donkin and Geranios very recently unified and extended these results to certain “framed staircase” partitions Reference DG18 which we will discuss (within the wider context of “2-separated” partitions) below. It is worth emphasising that for all Specht modules are indecomposable and therefore questions of decomposability and (non-simple) semisimplicity are inherently 2-modular problems. ,

For we show that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a sum of graded simple modules. Our proof makes heavy use of recent results in the ,*graded* representation theory of Hecke and rational Cherednik algebras. We shall denote the quantisations of the Specht and simple modules by and respectively over We completely determine the rows of the graded decomposition matrix of . labelled by 2-separated partitions; this serves as a first approximation to our goal and subsumes and generalises the results on decomposability and decomposition numbers of Specht modules for hook partitions (belonging to blocks of small -core)Reference Spe14Reference CMT04, and results on decomposition numbers of Specht modules in blocks of enormous 2-cores Reference JM96.

### Graded decomposition numbers of semisimple Specht modules

The partitions of interest to us (for both Saxl’s conjecture and our decomposability classification) are the 2-separated partitions. Such partitions are obtained by taking a staircase partition, and adding 2 copies of a partition , to the right of and 2 copies of a partition to the bottom of in such a way that and do not touch (except perhaps diagonally). Such partitions, denoted can be pictured as in Figure ,1.

Notice that if the weight of a block is small compared to the size of the core, then all partitions in that block are 2-separated. We emphasise that the size of the staircase in the following statement is immaterial (provided that where , denotes the length of the partition and so we simply write ), For those interested in the extra graded structure, we refer the reader to the full statement in Corollary .4.2.

In particular, there exist many blocks of (those with large cores) for which all Specht modules in the block are semisimple. In Reference DF12 Dodge and Fayers remark that *“every known example of a decomposable Specht module is labelled by a 2-separated partition”* and *“it is interesting to speculate whether the 2-separated condition is necessary for a Specht module to be decomposable”*. In fact in Section 6 we show that their speculation is *not* true by exhibiting two infinite families of decomposable Specht modules obtained by “inflating” the smallest decomposable Specht module (indexed by ).

Theorem A implies that all known examples of decomposable Specht modules for are obtained by reduction modulo from decomposable semisimple Specht modules for .

### Applications to Kronecker coefficients

We now discuss the results and insights which 2-modular representation theory affords us in the study of Kronecker coefficients. We verify the positivity of the Kronecker coefficients in Saxl’s conjecture for a large new class of partitions, and propose conjectural strengthened and generalised versions of Saxl’s original conjecture. Our first main theorem on Kronecker coefficients is as follows:

We now shift focus to the Kronecker coefficients labelled by 2-separated partitions. In what follows, we shall write for the Kronecker coefficient labelled by a staircase of size for some and some 2-separated partition of in other words, we do not encumber the notation by explicitly recording the size of the staircases involved. ;

We do not recall the definition of a pair here, but rather discuss some examples and consequences of Theorem -Carter–SaxlB. In particular, Theorem B implies that every 2-block contains a wealth of constituents of the Saxl square which can be deduced using our techniques. Carter–Saxl pairs cut across hook partitions, partitions of arbitrarily large Durfee size, symmetric and non-symmetric partitions, partitions from arbitrary blocks, and across the full range of the dominance order. (In fact, the only common trait of these partitions is that they label semisimple Specht modules for We shall illustrate below that the property of being a Carter–Saxl pair is actually very easy to work with diagrammatically. For example, the above theorem includes the infinite family of “framed staircases” as some of the simplest examples: these are partitions which interpolate hooks and staircases. More explicitly, these are the partitions of the form .) These can be pictured as in Figure .3 below.

We wish to provide bounds on the Kronecker coefficients: the maximal possible values obtained by Kronecker products are studied in Reference PPV16, and the Kronecker products whose coefficients are all as small as possible (namely all 0 or 1) are classified in Reference BB17. For constituents to partitions of depth at most 4, explicit formulae for their multiplicity in squares were provided by Saxl in 1987, and later work by Zisser and Vallejo, respectively. For the Kronecker coefficients studied here, the easiest (and well known) non-trivial case is so the Kronecker coefficients are even unbounded; this also holds for the other families corresponding to partitions of small depth. Lower bounds coming from character values on a specific class were obtained by Pak and Panova in ,Reference PP17, where also the asymptotic behaviour of the multiplicity of special constituents is studied. Theorem B allows us to provide explicit lower bounds on the Kronecker coefficients for new infinite families of Saxl constituents, where again the multiplicities are unbounded.

We now provide some examples of more complicated Carter–Saxl pairs. For if we first focus on the (unique) block of weight , we find 7 constituents in this block labelled by framed staircases as well as the Carter–Saxl pairs given (up to conjugation) in Figure 4 below.

Finally, we propose two extensions of Saxl’s conjecture based on its modular representation theoretic interpretation. The first conjecture reduces the problem to the case of partitions, but at the expense of working in the more difficult modular setting. We remark that towards Saxl’s conjecture over -regular it has already been verified that for any 2-regular partition , of the Kronecker coefficient is positive Reference Ike15, and so it is natural to hope that this can be extended to positive characteristic.

What could be a suitable candidate for arbitrary not just triangular numbers? ,

While this sounds reasonable, in fact, for larger it hardly restricts the search for a good candidate as almost any partition of is then a for some -core So as a guide towards finding a simple module . whose tensor square contains all simples, one would try to find a suitable symmetric for a small prime -core .

## 1. The Hecke algebra

Let be a commutative integral domain. We let denote the symmetric group on letters, with presentation

We are interested in the representation theory (over of symmetric groups and their deformations. Given ) we define the Hecke algebra , to be the unital associative with generators -algebra , …, , and relations

for We let . be the smallest integer such that or set if no such integer exists. If is a field of characteristic and , then is isomorphic to .

We define a composition, of , to be a finite sequence of non-negative integers whose sum, equals , If the sequence . is weakly decreasing, we say that is a partition; we denote the set of all partitions of by The number of non-zero parts of a partition, . is called its ,length, the size of the largest part is called the width, ; Given . its ,Young diagram is defined to be the configuration of nodes,

The conjugate partition, is the partition obtained by interchanging the rows and columns of , when ; the partition , is said to be symmetric. Given a node we define the content to be and the ( -)residue to be the value of modulo We now recall the .dominance ordering on partitions. Let be partitions. We write if

If and we write For . partitions such that we define the ,skew diagram, denoted to be the set difference between the Young diagrams of , and .

Given we define a ,tableau of shape to be a filling of the nodes of the Young diagram of with the numbers We define a .standard tableau to be a tableau in which the entries increase along both the rows and columns of each component. We let denote the set of all standard tableaux of shape We extend this to (standard) skew tableaux of shape . in the obvious fashion. Given we set , Given . we let , be the subtableau of whose entries belong to the set We write . if for all and refer to this as the dominance order on .

We let and denote the most and least dominant tableaux respectively. We let be the permutation such that For example, . and

### 1.1. Modular representation theory

Let be a field and The group algebra of the symmetric group . is a semisimple algebra if and only if is a field of characteristic By a result of Dipper and James, the Hecke algebra of the symmetric group is a non-semisimple algebra if . is a primitive root of unity for some th or and is a field of characteristic We shall now recall the basics of the non-semisimple representation theory of these algebras. .

Modular representation theory seeks to deconstruct the non-semisimple representations of an algebra in terms of their simple constituents. To this end, we define the *radical* of a finite-dimensional -module denoted , to be the smallest submodule of , such that the corresponding quotient is semisimple. We then let and inductively define the *radical series*, of , by We have a finite chain .

In the non-semisimple case, the Specht modules are no longer simple but they continue to play an important role in the representation theory of as we shall now see. We say that a partition is if there is no -regular such that We let . denote the set of all partitions of -regular Occasionally, we will also use the notation . in place of For . an arbitrary field, we have that

provides a full set of non-isomorphic simple Of course, the radical of a Specht module is not easy to compute! The passage between the Specht and simple modules is recorded in the -modules.decomposition matrix,

where denotes the multiplicity of as a composition factor of This matrix is uni-triangular with respect to the dominance ordering on . We have already seen in equation .Equation 1.1 that every column of the decomposition matrix contains an entry equal to 1; namely if then We now recall James’ regularisation theorem, which states that every row of the decomposition matrix contains an entry equal to 1 (and identifies this entry). .

We define the ( -)ladder number of a node to be The . thladder of is defined to be the set

The of -regularisation is the partition obtained by moving all of the nodes of as high along their ladders as possible. When each ladder of , is a complete north-east to south-westerly diagonal in In particular, when . the partition is obtained from by sliding nodes as high along their south-west to north-easterly diagonals as possible.

### 1.2. Brauer–Humphrey’s reciprocity

Given an partition and -regular the corresponding simple we let -module, denote its projective cover. Brauer–Humphrey’s reciprocity states that has a Specht filtration,

such that for each we have , for some dependent on and such that the multiplicity, in this filtration is given by ,

In other words, the column of the decomposition matrix determines the multiplicities in a Specht filtration of th This will be a key observation for our applications to Kronecker coefficients in Section .5.

### 1.3. 2-blocks

We first recall the block-structure of Hecke algebras in (quantum) characteristic (which will be the main case of interest in this paper). Throughout this section and can be taken to be an arbitrary field (although we are mainly interested in the cases when or is of characteristic The algebra ). decomposes as a direct sum of primitive 2-sided ideals, called blocks. All questions concerning modular representation theory break-down block-by-block according to this decomposition: in particular each simple/Specht module belongs to a unique block.

The rim of the Young diagram of is the collection of nodes Given . we define the associated ,rim-hook to be the set of nodes If . then we refer to , as a removable if -hook; we refer to as a removable domino. Removing from gives the Young diagram of a partition of It is easy to see that a partition has no removable dominoes if and only if it is of the form . for some in which case we say that it is a ,2-core. We let denote the 2-core partition obtained by successively removing all removable dominoes from (this defines a unique partition). The number of dominoes removed from is referred to as the weight of the partition and is denoted Given . we define ,