Supercharacters of unipotent groups defined by involutions
Abstract.
We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic matrices and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table.
Key words and phrases:
supercharacter, unipotent group2010 Mathematics Subject Classification:
20C33,05E101. Introduction
For a power of a prime, let denote the group of unipotent upper triangular matrices over the finite field with elements. Classifying the irreducible representations of is known to be a “wild” problem (see [10]). In [2], André constructs a set of characters, referred to as “basic characters,” such that each irreducible character of occurs with nonzero multiplicity in exactly one basic character. These characters can be thought of as a coarser approximation of the irreducible characters of . Diaconis–Isaacs generalize the idea of a basic character to a “supercharacter” of an arbitrary finite group in [8]. They also construct supercharacter theories for all finite algebra groups , which are subgroups of such that is an algebra. In the case that , the constructions of André and of Diaconis–Isaacs produce the same supercharacter theory. The two constructions use different techniques; André constructs basic characters by inducing linear characters from certain subgroups of , whereas Diaconis–Isaacs utilize the twosided action of on the associative algebra of strictly upper triangular matrices.
André–Neto have modified André’s earlier construction to the unitriangular groups in types and in [3, 4, 5]. In this paper, we generalize these supercharacter theories in a manner analogous to the type construction of Diaconis–Isaacs. The construction in [3, 4, 5] uses the idea of a “basic subset of roots” to induce linear characters from certain subgroups of the full unitriangular group. Our construction instead utilizes actions of on the Lie algebras of the unitriangular groups in types and to define superclasses and supercharacters. One advantage of our method is that it works in situations where the idea of a basic subset of roots does not make sense, such as the case of the unipotent radical of a parabolic subgroup.
Aguiar et al. construct a Hopf algebra on the type supercharacters in [1] and show that this structure is isomorphic to the Hopf algebra of symmetric functions in noncommuting variables. In [6], Benedetti has constructed an analogous Hopf algebra on the superclass functions of type . Marberg describes the type and supercharacters in terms of type supercharacters in [14]. We hope that our construction will allow for many more type results to be generalized to other types.
Given a pattern subgroup of (an algebra group such that has a basis of elementary matrices) and a subgroup of defined by an antiinvolution of , we construct a supercharacter theory. The antiinvolution of induces an action of on the Lie algebra of , which we use to construct the superclasses and supercharacters. The examples that naturally fall into this context include the unipotent orthogonal, symplectic, and unitary groups. Let denote the matrix with ones on the antidiagonal and zeroes elsewhere; for a power of an odd prime, define
The groups are the unipotent groups of types and , and the groups are the unipotent groups of type . Note that these groups are each defined by an antiinvolution of ; our construction produces the supercharacter theories constructed by André–Neto in [3, 4, 5].
We can also construct supercharacter theories of the unipotent unitary groups. For , define by . Let
The group is the group of unipotent unitary matrices over . As is a subgroup of that is defined by an antiinvolution, we get a supercharacter theory from the action of on the Lie algebra of . The supercharacter values on superclasses demonstrate Ennola duality, as they are obtained from the supercharacter values of by formally replacing ‘’ with ‘’.
We present the main result of the paper in Section 2, which is applied in Section 3 to construct supercharacter theories of unipotent orthogonal and symplectic groups. We develop necessary background material on the interactions between groups, algebras and vector spaces in Section 4. We review the construction of supercharacter theories of algebra groups in Section 5, and prove our main result in Section 6. Finally, in Section 7, we construct supercharacter theories of the unipotent unitary groups and calculate the values of supercharacters on superclasses.
2. Main result
The main result of this paper is the construction of a supercharacter theory for certain subgroups of algebra groups that are defined by antiinvolutions. In this section we review the necessary background material on algebra groups and present the main result of the paper.
2.1. Supercharacter theories
The idea of a supercharacter theory of an arbitrary finite group was introduced by Diaconis–Isaacs in [8], and has been connected to a number of areas of mathematics. In [11], Hendrickson shows that the supercharacter theories of a finite group are in bijection with the central Schur rings over . Brumbaugh et al. construct certain exponential sums of interest in number theory (e.g., Gauss, Ramanujan, and Kloosterman sums) as supercharacters of abelian groups in [7]. For a more indepth treatment of supercharacters see [8]; we only address the basics that are necessary for our construction.
Let be a finite group, and suppose that is a partition of into unions of conjugacy classes and is a set of characters of . We say that the pair is a supercharacter theory of if

,

the characters are constant on the members of , and

each irreducible character of is a constituent of exactly one character in .
The characters are referred to as supercharacters and the sets are called superclasses.
2.2. Algebra groups and pattern subgroups
Let be a field and let be a nilpotent associative algebra over . The algebra group associated to is the set of formal sums
with multiplication defined by (see [12]). As is nilpotent, elements in have inverses given by
We will often write to indicate that is the algebra group associated to .
For example, if we define to be the group of upper triangular matrices over with ones on the diagonal and to be the algebra of upper triangular matrices over with zeroes on the diagonal, then is the algebra group associated to .
Let be a poset on that is a suborder of the usual linear order. In other words, has the properties that

if then ,

if and then , and

for all .
Corresponding to the poset are a pattern subgroup
and a pattern subalgebra
Note that is the algebra group corresponding to . For a more complete discussion of pattern subgroups, see [9].
In [8], Diaconis–Isaacs construct a supercharacter theory for an arbitrary finite algebra group . Note that acts on by left and right multiplication; there are corresponding actions of on the dual given by
where , , and . Let
and let be a nontrivial homomorphism. For and , define
and
Theorem 5.1 ([8]).
The partition of given by , along with the set of characters , form a supercharacter theory of . This supercharacter theory is independent of the choice of .
The supercharacter theory is independent of in that the sets and do not depend on . If a different is chosen, the will be permuted. In Section 5 we present a modified proof of this result as motivation for the proof of our main result.
2.3. Subgroups of algebra groups defined by antiinvolutions
For a power of a prime, let be a nilpotent associative algebra of finite dimension over . Define . We equip with a Lie algebra structure given by .
Let
be an involutive associative algebra antiautomorphism, and for define . Note that this makes an involutive antiautomorphism of .
Define
and
Note that is not an associative algebra, although it is closed under the Lie bracket.
For and , define . It is routine to check that this defines a linear action of on . The action restricts to an action of on , and for and , . We can also define a left action of on itself by . This action restricts to an action of on , and for and , .
The motivating examples of groups defined in this manner are the unipotent orthogonal, symplectic, and unitary groups in odd characteristic. For instance, if and , with odd, we can define an antiautomorphism
where is the matrix with ones on the antidiagonal and zeroes elsewhere. Then
and
The unipotent symplectic and unitary groups can be similarly described in terms of antiautomorphisms of the upper triangular matrices.
2.4. Springer morphisms
In order to utilize the Lie algebra structure of to study , we would like a bijection between and that preserves useful properties. In the case of an algebra group , we can use the map to relate to . In general, however, it is not the case that , so we need a variation on this map. André–Neto define a bijection from to in [3], however we require a map that is invariant under the adjoint action of .
Given an algebra group and a map as above, we define a Springer morphism to be a bijection such that

, and

there exist such that .
The dependence of these conditions on is implicit in that and are defined in terms of . Note that condition (2) gives that for any algebra subgroup , and also guarantees that will be invariant under the adjoint action of . We require that the coefficient of the term of be 1 for ease of computation; relaxing this condition would not have any effect on the resulting supercharacter theory.
Springer morphisms are introduced by Springer and Steinberg in [15] (III, 3.12) and are utilized by Kawanaka in [13]. Our definition of a Springer morphism is slightly modified from the original definition, but the examples given below are Springer morphisms in the original sense. The logarithm map
is perhaps the most natural choice of a Springer morphism, but is not defined in many characteristics. The map
is, however, a Springer morphism in all odd characteristics. We mention that this is a constant multiple of the map , which is often referred to as the Cayley map (see, for instance, [13]). The following lemma is easy to verify directly.
Lemma 2.1.
Let be a power of the prime , and let and be the maps defined above. Let be any algebra group, and let be any antiinvolution of . If for all , then is a Springer morphism. If is odd, then is a Springer morphism.
This lemma allows us to assume the existence of a Springer morphism if we are working in odd characteristic, which we will do for the remainder of the paper.
2.5. Main theorem
Let be a power of an odd prime, and let be a pattern subgroup of for some and . For , define . We consider as an algebra; let be an antiinvolution of such that for all . In other words, reflects the entries of elements of across the antidiagonal, up to a constant multiple. The antiautomorphisms that define the orthogonal, symplectic and unitary groups all have this property. Let
and
Let be any Springer morphism and let be a nontrivial homomorphism. For , and , let and . For and , define
(2.1) 
and
(2.2) 
where is a constant determined by (and independent of the choice of as orbit representative). As in [8], can be written in terms of the sizes of orbits of group actions. If we let be the subgroup of defined by
then
(2.3) 
Theorem 6.1.
The partition of given by , along with the set of characters , form a supercharacter theory of . This supercharacter theory is independent of the choice of and .
The supercharacter theory is independent of and in that the sets and do not depend on these functions. If a different is chosen or condition (2) in the definition of a Springer morphism is relaxed to allow for other coefficients, the will be permuted. The supercharacter theory is also independent of the choice of subfield of ; that is, if is any subfield of and is an antiautomorphism of when viewed as an algebra, we get the same supercharacter theory as by considering as an algebra. We will prove this theorem in Section 6, along with the following result that allows us to relate our supercharacter theories to those of André–Neto.
Theorem 6.10.
The superclasses of are exactly the sets of the form , where is some superclass of .
Remark.
Note that the superclasses of are determined by an action of on (with one acting on each side), whereas the superclasses of only require a left action of on . This may seem strange, especially in light of Theorem 6.10. The reason that we only need one copy of to act on is due to the fact that if and , then there exists with . In other words, because the elements of respect an involution we only need one copy of acting on the left to construct the superclasses. For the details, see the proof of Theorem 6.10.
3. Supercharacter theories of unipotent orthogonal and symplectic groups
Before we prove Theorem 6.1, we use it in this section to construct supercharacter theories for two families of groups.
3.1. Supercharacter theories of unipotent orthogonal groups
Let be the matrix with ones on the antidiagonal and zeroes elsewhere, and let denote the transpose of a matrix . For a power of an odd prime, define
along with the corresponding Lie algebra
Define
Define an antiautomorphism of by . Note that satisfies the conditions required by Theorem 6.1, and furthermore
Define and as in 2.1 and 2.2 with and . By Theorem 6.1, there is a supercharacter theory of with superclasses and supercharacters .
In [4], André–Neto construct a supercharacter theory of . They show that their superclasses are the sets of the form , where is a superclass of under the algebra group supercharacter theory. In particular, the following theorem follows from Theorem 6.10.
Theorem 3.1.
The supercharacter theory of defined above coincides with that of André–Neto in [4].
We can also construct supercharacter theories of certain subgroups of using this method. We will call a poset a mirror poset if implies that (recall that ). The antiautomorphism as defined above restricts to an antiautomorphism of for any mirror poset. Furthermore,
Define and as in 2.1 and 2.2 with and . By Theorem 6.1, there is a supercharacter theory of with superclasses and supercharacters . By Theorem 6.10, the superclasses are of the form where is a superclass of in the algebra group supercharacter theory. In particular, if is the unipotent radical of a parabolic subgroup of then for some mirror poset .
There are two important examples of a subgroup obtained from a mirror poset in type . First, let be the mirror poset on defined by
Then , and we get a second supercharacter theory of which is at least as fine as the one originally defined. This new supercharacter theory is in fact strictly finer than the original; the elements and of are in the same orbit under the action of on , but in different orbits under the action of on .
We can also consider the poset on defined by
In this case, , and the supercharacter theory obtained is the algebra group supercharacter theory.
3.2. Supercharacter theories of unipotent symplectic groups
Define
where once again is the matrix with ones on the antidiagonal and zeroes elsewhere. For a power of an odd prime, define
along with the corresponding Lie algebra
Define
Define an antiautomorphism of by . Note that satisfies the conditions required by Theorem 6.1, and furthermore
Define and as in 2.1 and 2.2 with and . By Theorem 6.1, there is a supercharacter theory of with superclasses and supercharacters .
In [4], André–Neto have also constructed supercharacter theories of . As was the case with the unipotent orthogonal groups, the superclasses are the sets of the form , where is a superclass of under the algebra group supercharacter theory. In particular, the following theorem follows from Theorem 6.10.
Theorem 3.2.
The supercharacter theory of defined above coincides with that of André–Neto in [4].
We can also construct supercharacter theories of certain subgroups of just as we did for . The antiautomorphism as defined above restricts to an antiautomorphism of for any mirror poset. Furthermore,
Define and as in 2.1 and 2.2 with and . By Theorem 6.1, there is a supercharacter theory of with superclasses and supercharacters . By Theorem 6.10, the superclasses are of the form where is a superclass of in the algebra group supercharacter theory. In particular, if is the unipotent radical of a parabolic subgroup of then for some mirror poset .
4. Background
In order to prove Theorem 6.1 we need a number of lemmas with regards to the interactions between groups and vector spaces. In this section we will establish these results before applying them in Sections 5 and 6.
4.1. Linear actions of groups on vector spaces
Let be a finite group acting linearly on a finite dimensional vector space over a finite field. There is a corresponding linear action on the dual space ; for , , and , define
The following lemma relating the number of orbits of these two actions appears in [8].
Lemma 4.1 ([8], Lemma 4.1).
The actions of on and have the same number of orbits.
4.2. Complexvalued functions of certain groups
Let be a finite group, and let be a vector space over the finite field such that there exists a bijection . Let be a nontrivial linear character. We can use the vector space structure of to study the space of functions from to . The following lemma is a consequence of Lemma 5.1 in [8].
Lemma 4.2.
Let , and be as above.

[label=()]

The set of functions , where , form an orthonormal basis for the space of functions from to .

The set of functions , where , form an orthonormal basis for the space of functions from to .
The next lemma will be useful in describing certain induced characters.
Lemma 4.3.
Let be a vector space of finite dimension over with subspace , and let . Then
Proof.
Let be a subspace of such that . Let , and write , where and . Then
Observe that the set of functionals such that is exactly . Furthermore, for ,
as is nontrivial. ∎
Corollary 4.4.
If , then
is the regular character of .
The groups we are studying in this paper are all naturally in bijection with a vector space. We can consider an algebra group along with the bijection
We can also take our group to be as defined in Section 2.3, along with the corresponding Lie algebra and a Springer morphism . In these two cases, we can use the adjoint action of the group on its Lie algebra to understand certain induced representations.
Lemma 4.5.
Suppose that is a finite group, is a vector space over , and is a bijection. Suppose further that there is an action
such that for all . If is a subgroup of such that is a subspace of , and is a functional such that is a class function of , then
Proof.
5. Supercharacter theories of algebra groups
Let be an algebra group over the field , where is a power of a prime. Diaconis–Isaacs construct a supercharacter theory of in [8], which we describe here. Define
Note that acts by left multiplication, right multiplication, and conjugation on (with corresponding actions on ). For , define
Let be a nontrivial homomorphism. For , define
Note that the set is the orbit of under the action of on defined by . In particular, is the orbit of under the action of the normal subgroup . It follows that for all , and the definition of is independent of the choice of representative of .
Theorem 5.1 ([8]).
Let and be as above.

[label=()]

The functions are characters of .

The partition of given by , along with the set of characters , form a supercharacter theory of .
We present a proof of this result as motivation for our proof of Theorem 6.1; our method is different from that in [8], although many of the ideas are similar. We will prove (a) by proving a more specific result given in Theorem 5.4. Assuming (a), we have the following.
Proof of (b)..
We need to show that conditions (1)(3) in the definition of a supercharacter theory (see Section 2.1) are satisfied. For (1), note that is the number of orbits of the action of on defined by . At the same time, is the number of orbits of the corresponding action of on . By Lemma 4.1, the number of orbits of the two actions are equal.
To demonstrate that condition (2) holds, choose and ; we have that
It follows that only depends on the superclass of .
It remains to prove (a). Define
and let . It is worth mentioning that our notation differs from that of Diaconis–Isaacs. We define as above so that is the left ideal of that corresponds to the left orbit as follows.
Lemma 5.2 (Lemma 4.2 (d), [8]).
With notation as above, .
Diaconis–Isaacs prove the following result as part of Theorem 5.4 in [8].
Lemma 5.3.
The function is a linear character of .
Proof.
Let ; then
∎
We can now prove that the functions are characters of .
Theorem 5.4 ([8]).
With as defined above,
6. Supercharacter theories of unipotent groups defined by antiinvolutions
In this section we construct supercharacter theories of the groups that were introduced in Section 2. Let be a power of an odd prime, and let be a pattern subgroup of for some and . We consider as an algebra; let be an antiinvolution of such that for all (recall that ). Define
and
Let be any Springer morphism and let be a nontrivial homomorphism. Recall that there are left actions of and on defined by
for and , along with corresponding actions on .
In the construction of the supercharacter theories of algebra groups, the normal subgroup of plays an important role. We need an analogous subgroup of to construct a supercharacter theory of . Let
and define . Note that is a twosided ideal of , hence is a normal subgroup of .
For and , define
and
As is a normal subgroup of , is independent of the choice of orbit representative of .
Theorem 6.1.
Let and be as above.

[label=()]

The functions are characters of .

The partition of given by , along with the set of characters , form a supercharacter theory of .
We will prove (a) by proving a more specific result given by Theorem 6.9. Assuming (a), we have the following.
Proof of (b).
We need to show that conditions (1)(3) in the definition of a supercharacter theory (see Section 2.1) are satisfied. For condition (1), note that is the number of orbits of the action of on . At the same time, is the number of orbits of the corresponding action of on . By Lemma 4.1, the number of orbits of the two actions are equal.
To demonstrate that condition (2) holds, choose