# Coherent systems of probability measures on graphs for representations of free Frobenius towers

Henry Kvinge
###### Abstract.

First formally defined by Borodin and Olshanski, a coherent system on a graded graph is a sequence of probability measures which respect the action of certain down/up transition functions between graded components. In one common example of such a construction, each measure is the Plancherel measure for the symmetric group and the down transition function is induced from the inclusions .

In this paper we generalize the above framework to the case where is any free Frobenius tower and is no longer assumed to be semisimple. In particular, we describe two coherent systems on graded graphs defined by the representation theory of and connect one of these systems to a family of central elements of . When the algebras are not semisimple, the resulting coherent systems reflect the duality between simple -modules and indecomposable projective -modules.

## 1. Introduction

Study of the asymptotic representation theory of symmetric groups has uncovered deep results through the synthesis of techniques from probability theory, combinatorics, and algebra. One idea frequently used both implicitly and explicitly in this field is that of a coherent system. Coherent systems were first formally defined in [4] where they are a key part of the framework used to construct infinite-diffusion processes. Given a graded graph with vertex set , a coherent system consists of a sequence of probability measures on the graded components of , which are consistent with the action of a down transition function between components.

A basic example of this construction occurs when is taken to be the Plancherel measure on . The sequence is coherent with respect to down and up transition functions each controlled by the number of standard Young tableau of appropriate shape. On the other hand, this coherent system can also be described in terms of the representation theory of the tower of symmetric group algebras . From this point of view, is defined via the decomposition of the left regular representation of into a direct sum of simple representations, and and are defined via induction and restriction functors.

One of the main goals of this paper is to understand what assumptions are necessary to generalize the above construction to an arbitrary tower of algebras . We show that when for all :

1. the induction and restriction functors between representations and representations are biadjoint (so that is a Frobenius extension of ),

2. is a free -bimodule,

then there a two coherent systems that generalize the Plancherel system corresponding to . The observation that there are two such systems to choose from reflects the fact that when is not semisimple, there arises a distinction between simple -modules and indecomposable projective -modules. In particular, in one coherent system the down transition functions are controlled by the dimensions of simple -modules and the up transition functions are controlled by the dimensions of indecomposable projective -modules, while in the other coherent system the reverse is true.

That the existence of coherent systems should depend on categorical conditions is not surprising given the recent appearance of coherent systems within categorical representation theory. It was shown in [16] for example, that certain moments associated to down/up transition functions for the tower appear naturally within the center of Khovanov’s Heisenberg category [14], a monoidal category which conjecturally categorifies the infinite dimensional Heisenberg algebra. A similar phenomenon [17] was observed for down/up transition functions associated to towers of Sergeev algebras, which appear in the twisted Heisenberg category of Cautis and Sussan [7].

In the classical case of , Biane observed that data related to is encoded by elements of [2]. This allows for certain questions related to asymptotic processes to be translated into algebraic/combinatorial questions. We show that when each in the free Frobenius tower is further assumed to be a symmetric Frobenius algebra, then there are families of elements in which encode data not only for , but also . We plan to investigate these elements and their possible connection to symmetric functions in greater detail in a later paper.

One of the broader goals of this work is to pave the way for studies of the asymptotic representation theory of more exotic towers of algebras. In particular, because of their connections to Lie theory and geometry, we think that understanding the asymptotic representation theory of the wreath product algebras attached to a Frobenius algebra would be particularly interesting (see Example 2.1.3). Degenerate cyclotomic Hecke algebras are another natural generalization of symmetric groups (see Example 2.1.4). This family of non-semisimple towers of algebras has deep connections to Lie theory [15, Part 1]. Furthermore, as their representation theory is indexed by certain multipartitions, we hypothesize that using the coherent systems described in this paper, an entire new family of limit shapes could be constructed.

The paper is structured as follows: in Section 2 we review Frobenius extensions and Frobenius towers. In Section 3 we describe the representation theory of Frobenius towers particularly in relation to elements from the centralizers . In Section 4 we define a pair of coherent systems on certain graphs attached to free Frobenius towers and connect one of these to a family of central elements. Finally in Section 4.1 we discuss future directions of research.

Acknowledgements: The idea for this paper arose from extended discussions with Anthony Licata. The author would like to thank him for these and for continued feedback during the course of this paper’s development. Without his help, it is unlikely that this paper would have been written. The author would also like to thank Alistair Savage, Monica Vazirani, Leonid Petrov, and Ben Webster for helpful discussions.

## 2. Frobenius extensions and Frobenius towers

In this section we review some of the basic properties of Frobenius extensions and Frobenius towers. All the algebras that we introduce are assumed to be finite-dimensional -algebras unless otherwise stated. We point the reader toward [12] for a detailed introduction to Frobenius extensions as well as a more diverse set of examples.

Recall that a tower of -algebras is a nested sequence of algebras

 A0⊂A1⊂A2⊂… (1)

In this paper we will always assume that which adds nice properties to the tower. Due to the inclusions in (1), for any , is an -bimodule, with the action of given by the natural left-multiplication against and the action of given by the natural right-multiplication against . When considering as a bimodule we will sometimes use the following notation:

• To denote as a -bimodule we write .

• To denote as a -bimodule we write .

• To denote as a -bimodule we write .

• To denote as a -bimodule we write .

This bimodule structure defines a restriction functor and an induction functor for each , where (resp. ) is the category of finite-dimensional -modules (resp. -modules). Specifically for and ,

 N\resAnAk−−−−→k(An)⊗AnN. (2)

and

 K\indAnAk−−−−→(An)k⊗AkK. (3)

Henceforth for brevity we write

 \indnk:=\indAnAkand\resnk:=\resAnAk

when the tower of algebras is understood from the context. Via (2) we can identify with the -bimodule and via (3) we can identify with the -bimodule . This description allows us to translate natural transformations between compositions of the functors and into bimodule homomorphisms between tensor products of and .

###### Example 2.1.
1. The tower of symmetric group algebras is our motivating example for this paper. Recall that is generated by Coxeter generators subject to relations

 s2i=1, 1≤i≤n−1, (4) sisj=sjsi, |i−j|>1,1≤i,j≤n−1, (5) sisi+1si=si+1sisi+1, 1≤i≤n−2. (6)

We will always assume that the inclusion is the standard one with mapping to the subgroup of generated by . By linear extension this defines an inclusion .

2. Recall that the Clifford superalgebra is the unital associative algebra with generators such that for :

 c2i=1andcicj=−cjci for i≠j.

The superalgebra structure is defined by setting each generator to be an odd element. The Sergeev superalgebra is defined as , where the action of is given by permuting the indices of so that:

 sici=ci+1si, 1≤i≤n−1, sici+1=cisi, 1≤i≤n−1, sicj=cjsi j≠i,i+1.

The Sergeev algebras form a tower via the embedding

 \MBSn↪\MBSn+1

which sends for and for .

One reason that Sergeev algebras are interesting is that studying simple representations of is equivalent to studying projective representations of . See [15] and [24] for detailed studies of the algebras , their representation theory, and connections to combinatorics and Lie theory. is another tower of algebras whose asymptotic representation theory has been well-studied, see for example [9], [10], [3], [21].

3. Let be a Frobenius graded superalgebra (that is, is graded). The symmetric group acts on by superpermutations. More precisely, for homogeneous and ,

 si⋅(f1⊗⋯⊗fi⊗fi+1⊗⋯⊗fn) (7)
 =(−1)¯fi\widebarfi+1(f1⊗⋯⊗fi+1⊗fi⊗⋯⊗fn)

where and denote the -degree of and respectively.

The wreath product algebras induced from (7) form a tower via the embedding

 F⊗n⋊\Syn↪F⊗(n+1)⋊\Syn+1

which for and sends

 (f1⊗⋯⊗fn)ω↦((f1⊗⋯⊗fn⊗1)ω.

also inherits a -grading from by setting to sit in degree .

In fact, Example 2.1.2 is a special case of with . For a description of the representation theory of in the non-super, ungraded setting see [19, Theorem A.5], while for the super, graded case see [22, Section 4].

4. The degenerate affine Hecke algebra is generated by elements and , such that satisfy relations (4)-(6), commute, and:

 sjxi=xisj, i≠j,j+1, sixi=xi+1si−1, 1≤i≤n−1.

Thus as a -vector space

 Hn≅\MBC[x1,…,xn]⊗\MBC[\Syn].

Let be the dominant weight lattice for and let be the associated Dynkin indexing set. Also let where and is the th fundamental weight. The integer is called the level of . Let be the two-sided ideal of generated by the element

 ∏i∈I(x1−i)λi.

The quotient algebra is called the degenerate cyclotomic Hecke algebra associated to . By abuse of notation we write for the images of in this quotient. is finite dimensional, with the set

 {xt11xt22…xtnnσ|t1,t2,…,tn

being a basis [15, Theorem 3.2.2], so that . The algebras generalize symmetric group algebras because when , .

The collection forms a tower via the embeddings

 Hλn↪Hλn+1

which send for and for . The tower has a rich representation theory which is described in detail in [15, Part I]. In particular it has a crystal structure isomorphic to the highest weight crystal in affine type . The algebras will be one of our prime examples of a tower of non-semisimple algebras.

In all the examples above, we follow the usual convention that .

###### Remark 2.2.

It is worth noting that all the examples in Example 2.1 are particular instances of cyclotomic wreath product algebras as defined in [23]. While we could have thus presented a single unified example, we felt it was more appropriate to partition our examples into digestible chunks as above for the benefit of the reader that may not be familiar with cyclotomic wreath product algebras.

For it is always the case that the functor is left adjoint to . This is known as Frobenius reciprocity. When is also left adjoint to , so that are a pair of biadjoint functors, then we say that is a Frobenius extension of . When in addition is a free -bimodule, then we say that is a free Frobenius extension of . If is a Frobenius extension of for all in the tower , then we say that is a Frobenius tower. When is a free Frobenius extension of for all , then we say that is a free Frobenius tower.

###### Theorem 2.3.

[1, Corollary 1.2] The algebra is a free Frobenius extension of if and only if there is a -bimodule homomorphism

 En+1,n:n(An+1)n→An,

a finite set of elements , and a vector space isomorphism such that for any ,

 a=∑b∈Bn+1,nEn+1,n(ab∨)b=∑b∈Bn+1,nb∨En+1,n(ba). (8)

We call and a dual basis pair, a Frobenius homomorphism, and the triple a Frobenius system.

###### Corollary 2.4.

When is a free Frobenius extension of then is a basis for as a left -module, is a basis for as a right -module, and

 En+1,n(b(b′)∨)=δb,b′for any b,b′∈Bn+1,n.
###### Example 2.5.

All the towers listed in Example 2.1 are examples of free Frobenius towers. We describe their Frobenius homomorphisms and examples of dual bases below.

1. For and , is defined on group elements so that for

 En+1,n(g)={gif g∈\Syn0otherwise.

Alternatively, since as an -bimodule decomposes as

 n(\MBC[\Syn+1])n≅\MBC[\Syn]⊕(\MBC[\Syn]⊗\MBC[\Syn−1]\MBC[\Syn]), (9)

then can also be viewed as projection onto the first summand on the right side of (9).

One choice for is the set of right coset representatives of in . Then is the set of left coset representatives of in . In particular, one dual basis pair is

 Bn+1,n={snsn−1…si|1≤i≤n+1}

and

 B∨n+1,n={si…sn−1sn|1≤i≤n+1}

(here, the convention is that when , the corresponding element is the identity). Note that in this case sends an element of to its inverse.

2. Let be a Frobenius graded superalgebra with trace map of degree . Let and be homogeneous dual bases of with respect to .

If , then from [22, Proposition 3.4], as an -bimodule decomposes as

 n(An+1)n≅(⨁b∈BF(1⊗⋯⊗1⊗b)An)⊕(An⊗An−1An)

and

 (1⊗⋯⊗1⊗b)An≅Π¯bAn,

where is the parity shift functor (recall that is the parity of in the -grading). Then corresponds to the -bimodule homomorphism that first projects

 An+1↠⨁b∈BF(1⊗⋯⊗1⊗b)An

followed by an -bimodule homomorphism

 ⨁b∈BF(1⊗⋯⊗1⊗b)An↠An.

From [22, Proposition 3.4], for and ,

 En+1,n((f1⊗⋯⊗fn+1)σ) ={(−1)τ(\widebarf1+⋯+\widebarfn)\tr(fn+1)(f1⊗⋯⊗fn)σif σ∈\Syn0otherwise.

Corresponding dual bases are

 Bn+1,n={(1⊗n⊗b)sn…si|b∈BF,1≤i≤n+1}

and

 B∨n+1,n={si…sn(1⊗n⊗b)|b∈ˆBF,1≤i≤n+1}.
3. For with of level , by [15, Lemma 7.6.1], as an -bimodule, decomposes as

 n(Hλn+1)n≅(d−1⨁j=0Hλnxjn+1)⊕(Hλn⊗Hλn−1Hλn)

where . Then corresponds to projection to the summand . Recall that in the case where then , and is the Frobenius homomorphism described in Example 2.5.1.

Define

 yn,k:=d−1∑t=k(−1)d−1−txt−kndet(En,n−1(xd+j−in))i,j=1,…,d−1−t

where the determinant in this expression is when . By [20, Proposition 5.11], a pair of dual bases with respect to is

 Bn+1,n={sn…siyi,a|i=1,…,n+1,a=0,…,d−1}

and

 B∨n+1,n={xaisi…sn|i=1,…,n+1,a=0,…,d−1}.

While we only required that the algebra immediately above be a Frobenius extension of , this assumption is in fact enough to ensure that for any , is a Frobenius extension of .

###### Proposition 2.6.

[12, Section 1.3] Let be a Frobenius tower. Then for any , is a Frobenius extension of with Frobenius homomorphism , and dual basis pair

 Bn,k:={b(k+1)…b(n−1)b(n)|b(i)∈Bi,i−1}

and

 B∨n,k:={(b(n))∨(b(n−1))∨…(b(k+1))∨|b(i)∈Bi,i−1}.
###### Remark 2.7.

As a consequence of the assumption that , each in the Frobenius tower is actually a Frobenius algebra with trace .

When for all ,

 En+1,k(aa′)=En+1,k(a′a)

then we say that is a symmetric Frobenius extension of . Note that when this property is true for , then is a symmetric Frobenius algebra (also known as a symmetric algebra). We will generally assume that each in our tower is symmetric as a Frobenius algebra (this is true for all towers in Example 2.5), though we do not assume that is symmetric for (this is not true for in Example 2.1.4).

The Frobenius homomorphism actually gives the co-unit of the adjunction of the functor and its right adjoint . That is, the natural transformation can be translated to an -bimodule homomorphism from to . This homomorphism is .

In the language of bimodules, the unit of the adjunction with the right adjoint of defines an -bimodule homomorphism . For , sends

 aHn,k∣−−−→∑b∈Bn,kab∨⊗b.

For any , defines an -bimodule homomorphism

 Tc,n,k:(An)k⊗Akk(An)→(An)k⊗Akk(An)

via the assignment such that for ,

 a⊗a′Tc,n,k∣−−−→ac⊗a′.

Finally, there is an -bimodule homomorphism , known as the multiplication map which maps to

 a⊗a′Mn,k∣−−−→aa′

(in fact this is the co-unit of the adjunction of with as its right adjoint).

We then define so that for ,

 c↦(Mn,k∘Tc,n,k∘Hn,k)(1)=∑b∈Bn,kb∨cb.
###### Remark 2.8.

When is the tower of Iwahori-Hecke algebras, the map is sometimes referred to as the relative norm [11].

### 2.1. The centers of {An}n≥0

We write for the center of and for the centralizer of in . That is

 Z(An,Ak):={a∈An|aa′=a′a,for all a′∈Ak}.

For all :

1. maps into .

2. maps into .

###### Proof.

Part (1) follows from the fact that is a -bimodule homomorphism. Let . Then for any ,

 a′En,k(a)=En,k(a′a)=En,k(aa′)=En,k(a)a′.

Part (2) follows from the observation that for any ,

 Nn,k(a)=∑b∈Bn,kb∨ab=(Mn,k∘Ta,n,k∘Hn,k)(1).

Since each map , , and is an -bimodule homomorphism, then for any ,

 a′(Mn,k∘Ta,n,k∘Hn,k)(1)=(Mn,k∘Ta,n,k∘Hn,k)(a′) =(Mn,k∘Ta,n,k∘Hn,k)(1)a′.

For , the elements will play a central role in this paper. We therefore set

 Cn,k:=Nn,k(1)=∑b∈Bn,kb∨b.

Note that for all , belongs to . In [12], the element is referred to as the -index of the Frobenius extension of . When and is symmetric, is referred to as the central Casimir element in [5].

###### Lemma 2.10.

[5, Proposition 3.5] When is a symmetric Frobenius algebra with respect to then

 Cn,0=∑b∈Bn,0b∨b=∑b∈Bn,0bb∨.

In general,

 ∑b∈Bn,kb∨b≠∑b∈Bn,kbb∨ (10)

(see Example 2.11.2 below). Because the element on the right side of (10) will appear again later, we denote it by

 Cιn,k:=∑b∈Bn,kbb∨.
###### Example 2.11.
1. When and then

 Cιn,k=Cn,k=∑g∈\Syn/\Sykgg−1=∣∣\Syn/\Syk∣∣=(n)(n−1)…(k+1).
2. Let for so that the level of is . Then with the help of Lemma 5.12 in [20] a routine calculation shows:

 C3,2=2(x1+x2+x3)−3(i+j)∈Hλ3.

On the other hand

 Cι3,2=2(s2s1x1s1s2+s2x2s2+x3)−3(i+j)
 =(6x3−4x3s2−2s2s1s2)−3(i+j)∈Hλ3.

It is not hard to show that in this case is not even in .

## 3. The representation theory of Frobenius towers

We begin by setting some notation related to the representation theory of and recalling some fundamental facts. Much of this section is modeled after the representation theory for degenerate cyclotomic Hecke algebras (Example 2.1.4) in [15, Part I]. In this section we assume that is a Frobenius tower. We denote the indexing set of isomorphism classes of simple -representations by (note that this is a finite set since is assumed to be finite-dimensional) and set

 Γ:=⋃n≥0Γn.

For let be the simple -module corresponding to . Each has a unique projective cover which we denote by . The set gives a complete list of isomorphism classes of indecomposable projective -modules. Recall from Section 2 that we write for the category of finite-dimensional -modules. We furthermore set to be the category of finite-dimensional projective -modules.

We denote by the triangulated Grothendieck group of finite dimensional -modules. That is, is the abelian group generated by the set subject to the relations such that for ,

 [M]+[N]=[K]

if there is a short exact sequence

 \zero→M→K→N→\zero.

is a free abelian group with basis . We denote by the split Grothendieck group of finite dimensional projective -modules, so that is the abelian group generated by with the property that for ,

 [P]+[P′]=[P′′]

if and only if

 P⊕P′≅P′′.

is a free abelian group with basis . We set

 G0(A)=⨁n≥0G0(An\Mod)andK0(A)=⨁n≥0K0(An\Pmod).

Note that since both and are exact functors that send projectives to projectives, they descend to linear operators on and . By abuse of notation we denote both the genuine functors and and their corresponding linear operators on and using the same notation.

There is a form