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Supercharacters of unipotent groups defined by involutions (extended abstract).

1 Introduction

For q a power of a prime, let [UT.sub.n] ([F.sub.q]) denote the group of unipotent n x n upper triangular matrices over the q element field [F.sub.q]. Classifying the irreducible representations of [UT.sub.n]([F.sub.q]) is known to be a "wild" problem (see [[GKP.sup.+]90]). In [And95, And02], Andre constructs a set of characters, referred to as "basic characters," such that each irreducible character of [UT.sub.n]([F.sub.q]) occurs with nonzero multiplicity in exactly one basic character. These characters can be thought of as a coarser approximation of the irreducible characters of [UT.sub.n]([F.sub.q]). Diaconis-Isaacs generalize the idea of a basic character to a "supercharacter" of an arbitrary finite group in [DI08]. They also construct supercharacter theories for all finite algebra groups G, which are subgroups of [UT.sub.n]([F.sub.q]) such that {g - 1 | g [member of] G} is an [F.sub.q]-algebra. In the case that G = [UT.sub.n]([F.sub.q]), the constructions of Andre and of Diaconis-Isaacs produce the same supercharacter theory. The two constructions use different techniques; Andre constructs basic characters by inducing linear characters from certain subgroups of [UT.sub.n]([F.sub.q]), whereas Diaconis-Isaacs utilize the two-sided action of [UT.sub.n]([F.sub.q]) on the associative algebra of strictly upper triangular matrices.

Andre-Neto have modified Andre's earlier construction to the unitriangular groups in types B, C and D in [AN06, AN09a, AN09b]. In this paper, we generalize these supercharacter theories in a manner analogous to the type A construction of Diaconis-Isaacs. The construction in [AN06, AN09a, AN09b] 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 [UT.sub.n]([F.sub.q]) on the Lie algebras of the unitriangular groups in types B, C and D 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.

Given a pattern subgroup G (an algebra group such that {g - 1 | g [member of] G} has a basis of elementary matrices) of the unipotent upper triangular matrices and a subgroup U of G defined by an anti-involution of G, we construct a supercharacter theory. The anti-involution of G induces an action of G on the Lie algebra of U, 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 J denote the n x n matrix with ones on the anti-diagonal and zeroes elsewhere, and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For q a power of an odd prime, define

[UO.sub.n]([F.sub.q]) = {g [member of] [UT.sub.n]([F.sub.q]) | [g.sup.-1] = J[g.sup.t]J} and

[US.sub.p2n]([F.sub.q]) = {g [member of] [UT.sub.2n]([F.sub.q]) | [g.sup.-1] = -U[g.sup.t][OMEGA]}.

The groups [UO.sub.n]([F.sub.q]) are the unipotent groups of types B and D, and the groups [US.sub.p2n]([F.sub.q]) are the unipotent groups of type C. Note that these groups are each defined by an anti-involution of [UT.sub.n] ([F.sub.q]); our construction produces the supercharacter theories constructed by Andre-Neto in [AN06, AN09a, AN09b].

We can also construct supercharacter theories of the unipotent unitary groups. For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], define [bar.g] by [([bar.g]).sub.ij] = [([g.sub.ij]).sup.q]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the group of unipotent unitary n x n matrices over [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a subgroup of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is defined by an anti-involution, we get a supercharacter theory from the action of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the Lie algebra of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The supercharacters and superclasses of the unipotent unitary group are indexed by labeled set partitions. This has been shown for types A, B, C and D in [And02, AN09a]. In Section 5, we describe this indexing, and calculate the supercharacter table of the unipotent unitary group in terms of labeled set partitions. The supercharacter table demonstrates Ennola duality, as the entries are obtained from the supercharacter table of [UT.sub.n]([F.sub.q]) by formally replacing 'q' with '-q'.

2 Preliminaries

2.1 Supercharacter theories

The idea of a supercharacter theory of an arbitrary finite group was introduced by Diaconis-Isaacs in [DI08]. Let G be a finite group and let Irr(G) denote the set of irreducible characters of G. For a subset X [subset or equal to] Irr(G), define

[[sigma].sub.X] = [summation over ([chi][member of]X)] [chi](1)[chi].

Suppose that K is a partition of G into unions of conjugacy classes and X is a partition of Irr(G). For each X [member of] X, choose a character [chi]X of G such that [chi]X = [a.sub.X][[sigma].sub.X] for some nonzero constant ax (in particular, one could choose [chi]X = [[sigma].sub.X]). We say that the partitions K and X, along with the characters xx, form a supercharacter theory of G if

1. [absolute value of X] = [absolute value of K],

2. the characters [chi]x are constant on the members of K, and

3. the set {1} [member of] K.

The characters [chi]X are referred to as supercharacters and the sets K [member of] K are called superclasses.

2.2 Labeled set partitions

A set partition v of [n] is a set of arcs i [??] j such that

1. 1 < i < j < n, and

2. if i [??] j, k [??] l [member of] v, then i = k if and only if j = l.

We will associate to a set partition a diagram consisting of nodes connected by arcs. For instance, if v = {1 [??] 2,2 [??] 6, 5 [??] 7} is a set partition of [7], then v corresponds to the diagram

By considering connected components of the diagram, we can associate v to a partition of the set [n] (in this case {{1,2, 6}, {3}, {4}, {5,7}}), which is the standard definition of a set partition.

The advantage of our definition of a set partition is that we can attach labels to arcs. If [F.sub.q] is the finite field with q elements, an [F.sub.q]-set partition of [n] is a set partition of [n] with arcs labeled by nonzero elements of [F.sub.q]. We will associate to an [F.sub.q]-set partition a diagram consisting of nodes connected by labeled arcs. For instance, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then v corresponds to the diagram

For a more complete discussion of labeled set partitions, see [Mar12].

2.3 Algebra groups and pattern subgroups

Let F be a field and let g be a nilpotent associative algebra over F. The algebra group G associated to g is the set of formal sums

G = {1 + x | x [member of] g}

with multiplication defined by (1 + x)(1 + y) = 1 + (x + y + xy) (see [Isa95]). As g is nilpotent, elements in G have inverses given by

[(1 + x).sup.-1] = 1 + [[infinity].summation over (i=1)] [(-x).sup.i].

We will often write G = 1 + g to indicate that G is the algebra group associated to g.

For example, if we define [UT.sub.n]([F.sub.q]) to be the group of n x n upper triangular matrices over [F.sub.q] with ones on the diagonal and [ut.sub.n]([F.sub.q]) to be the algebra of n x n upper triangular matrices over [F.sub.q] with zeroes on the diagonal, then [UT.sub.n]([F.sub.q]) is the algebra group associated to [ut.sub.n]([F.sub.q]). Let P be a poset on [n] which is a sub-order of the usual total linear order. Corresponding to P are a pattern subgroup

[U.sub.P] = {g [member of] [UT.sub.n]([F.sub.q]) | [g.sub.ij] = 0 unless i [[??].sub.P] j}

and a pattern subalgebra

[u.sub.P] = {x [member of] [ut.sub.n]([F.sub.q]) | [x.sub.ij] = 0 unless i [[??].sub.P] j}.

Note that [U.sub.P] is the algebra group corresponding to [u.sub.P]. For a more complete discussion of pattern subgroups, see [DT09].

In [DI08], Diaconis-Isaacs construct a supercharacter theory for an arbitrary finite algebra group G = 1 + g. Note that G acts on g by left and right multiplication; there are corresponding actions of G on the dual [g.sup.*] given by

(g[lambda])(x) = [lambda]([g.sup.-1] x) and ([lambda]g)(x) = [lambda](x[g.sup.-1]),

where g [member of] G, A [member of] [g.sup.*], and x [member of] g. Let

f: G [right arrow] g g [??] g - 1

and let [theta]: [F.sup.+.sub.q] [right arrow] [C.sup.x] be a nontrivial homomorphism. For g [member of] G and [lambda] [member of] [g.sup.*], define

[K.sub.g] = {h [member of] G | f(h) [member of] G/(g)G}

and

[[chi].sub.[lambda]] = [absolute value of G[lambda]/G[lambda]G] [summation over ([mu][member of]G[lambda]G)] [theta] x [mu] x f

Theorem 2.1 (([DI08])) The partition of G given by K = {[K.sub.g] | g [member of] G}, along with the set of characters {[[chi].sub.[lambda]] | [lambda] [member of] [g.sup.*]}, form a supercharacter theory of G. This supercharacter theory is independent of the choice of [theta].

The supercharacter theory is independent of [theta] in that the sets K and {[[chi].sub.[lambda]] | [alpha] [member of] [u.sup.*]} do not depend on 0. If a different [theta] is chosen, the [[chi].sub.[lambda]] will be permuted.

3 Main result

The main result of this paper is the construction of a supercharacter theory for certain subgroups of algebra groups which are defined by anti-involutions. In this section we define these groups and present the main result of the paper.

3.1 Subgroups of algebra groups defined by anti-involutions

For q a power of a prime, let g be a nilpotent associative algebra of finite dimension over [F.sub.q]. Let

[dagger]: g [right arrow] g x [??] [x.sup.[dagger]]

be an involutive associative algebra antiautomorphism, and for x [member of] g define [(1 + x).sup.[dagger]] = 1 + [x.sup.[dagger]]. Note that this makes t an involutive antiautomorphism of G. Define

U = {u [member of] G | [u.sup.[dagger]] = [u.sup.-1]}

and

u = {x [member of] g | [x.sup.[dagger]] = -x}.

Note that u is not an associative algebra, although it is closed under the Lie bracket. The motivating examples of groups defined in this manner are the unipotent orthogonal, symplectic, and unitary groups in odd characteristic.

3.2 Springer morphisms

In order to utilize the Lie algebra structure of u in studying U, we would like a bijection between U and u that preserves useful properties. In the case of an algebra group G, we can use the map g [??] g - 1 to relate G to g. In general, however, it is not the case that U = 1 + u, so we need a variation on this map. Andre-Neto define a bijection from U to u in [AN06], however we require a map that is invariant under the adjoint action of U.

Given an algebra group G = 1 + g and a map [dagger] as above, we define a Springer morphism f: G [??] g to be a bijection such that

1. f(U) = u and [f.sup.-1](u) = U.

2. There exist [a.sub.i] [member of] [F.sub.q] such that f (1 + x) = x + [[summation].sup.[infinity].sub.i=2] [a.sub.i][x.sup.i].

Springer morphisms are introduced by Springer and Steinberg in [SS70] (III, 3.12) and are utilized by Kawanaka in [Kaw85]. The following lemma is easy to verify directly.

Lemma 3.1 Let q be a power of an odd prime p, let G = 1 + g be any algebra group, and let [dagger] be any anti-involution of g. Then the map

f (1 + x) = 2x[(x + 2).sup.-1].

is a Springer morphism.

The map f is a constant multiple of the map 1 + x [??] x[(x + 2).sup.-1], which is often referred to as the Cayley map (see, for instance, [Kaw85]). 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.

3.3 Main theorem

Let q be a power of an odd prime, and let G = 1 + g be a pattern subgroup of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some n and k. For 1 [less than or equal to] i [less than or equal to] n, define [bar.i] = n +1 - i. We consider g as an [F.sub.q]- algebra; let [dagger] be an anti-involution of g such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In other words, f reflects the entries of elements of g across the antidiagonal, up to a constant multiple. The antiautomorphisms which define the orthogonal, symplectic and unitary groups all have this property. Let

U = {u [member of] G | [u.sup.[dagger]] = [u.sup.-1]}

and

u = {x [member of] g | [x.sup.[dagger]] = -x}

as above. Let f be any Springer morphism and let [theta]: [F.sup.+.sub.q] [right arrow] [C.sup.x] be a nontrivial homomorphism.

For g [member of] G, x [member of] u and [lambda] [member of] [u.sup.*], let g x x = gx[g.sup.[dagger]] and (g x [lambda])(x) = [lambda]([g.sup.-1] x x). For [lambda] [member of] [u.sup.*] and u [member of] U, define

[K.sub.u] = {[upsilon] [member of] U | f([upsilon]) [member of] G x f (u)} (1)

and

[[chi].sub.[lambda]] = 1/[n.sub.[lambda]] [summation over ([mu][member of]G x [kappa])] [theta] x [mu] x f, (2)

where [n.sub.[lambda]] is a constant determined by [lambda] (and independent of the choice of [lambda] as orbit representative). As in [DI08], [n.sub.[lambda]] can be written in terms of the sizes of orbits of group actions. If we let H be the subgroup of G defined by

H = {h [member of] G | [h.sub.ij] = 0 if j [less than or equal to] n/2},

then

[n.sub.[lambda]] = [absolute value of G x [lambda]]/[absolute value of H x [lambda]]. (3)

Theorem 3.2 The partition of U given by K = {[K.sub.u] | u [member of] U}, along with the set of characters {[[chi].sub.[lambda]]| [lambda] [member of] [u.sup.*]},form a supercharacter theory of U. This supercharacter theory is independent of the choice of 0 and f.

The supercharacter theory is independent of [theta] and f in that the sets K and {[[chi].sub.[lambda]] | [lambda] [member of] [u.sup.*]} do not depend on these functions. If a different [theta] is chosen or condition (2) in the definition of a Springer morphism is relaxed to allow for other x coefficients, the [[chi].sub.[lambda]] will be permuted. The supercharacter theory is also independent of the choice of subfield of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; that is, if F is any subfield of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [dagger] is an antiautomorphism of g when viewed as an F-algebra, we get the same supercharacter theory as by considering g as an [F.sub.q]-algebra. The following result allows us to relate our supercharacter theories to those of Andre and Neto.

Theorem 3.3 The superclasses of U are exactly the sets of the form U [intersection] [K.sub.g], where Kg is some superclass of G.

4 Supercharacter theories of unipotent orthogonal and symplectic groups

In this section we use Theorem 3.2 to construct supercharacter theories for two families of groups.

4.1 Supercharacter theories of unipotent orthogonal groups

Let J be the n x n matrix with ones on the antidiagonal and zeroes elsewhere, and let [x.sup.t] denote the transpose of a matrix x. For q a power of an odd prime, define

[O.sub.n]([F.sub.q]) = {g [member of] [GL.sub.n]([F.sub.q]) | [g.sup.-1] = J[g.sup.t]J}

along with the corresponding Lie algebra

[o.sub.n]([F.sub.q]) = {x [member of] [gl.sub.n]([F.sub.q]) | -x = J[x.sup.t]J}.

Let [UT.sub.n]([F.sub.q]) be the set of unipotent upper triangular matrices of [GL.sub.n]([F.sub.q]) and [ut.sub.n]([F.sub.q]) be the set of strictly upper triangular matrices of [gl.sub.n]([F.sub.q]). Define

U[O.sub.n]([F.sub.q]) = [UT.sub.n]([F.sub.q]) [intersection] [O.sub.n]([F.sub.q]) and u[o.sub.n]([F.sub.q]) = [ut.sub.n]([F.sub.q]) [intersection] [o.sub.n]([F.sub.q]).

Define an antiautomorphism [dagger] of [ut.sub.n]([F.sub.q]) by [x.sup.[dagger]] = J[x.sup.t]J. Note that [dagger] satisfies the conditions required by Theorem 3.2, and furthermore

U[O.sub.n]([F.sub.q]) = {g [member of] [UT.sub.2n]([F.sub.q]) | [g.sup.-1] = [g.sup.[dagger]]} and

[uo.sub.n]([F.sub.q]) = {x [member of] [Ut.sub.2n]([F.sub.q]) | -x = [x.sup.[dagger]]}.

Define [K.sub.u] and [[chi].sub.[lambda]] as in 1 and 2 with U = U[O.sub.n]([F.sub.q]) and u = [uo.sub.n]([F.sub.q]). By Theorem 3.2, there is a supercharacter theory of U[O.sub.n]([F.sub.q]) with superclasses {[K.sub.u]} and supercharacters {[[chi].sub.[lambda]]}.

In [AN09a], Andre-Neto construct a supercharacter theory of U[O.sub.n]([F.sub.q]). They show that their superclasses are the sets of the form U[O.sub.n]([F.sub.q]) [intersection] [K.sub.g], where [K.sub.g] is a superclass of [UT.sub.n]([F.sub.q]) under the algebra group supercharacter theory. In particular, the following theorem follows from Theorem 3.3.

Theorem 4.1 The supercharacter theory of U[O.sub.n]([F.sub.q]) defined above coincides with that of Andre-Neto in [AN09a].

We can also construct supercharacter theories of certain subgroups of U[O.sub.n]([F.sub.q]) using this method. We will call a poset P symmetric if i [[??].sub.P] j implies that j [[??].sub.P] [bar.i] (recall that [bar.i] = n - i + 1). The antiautomorphism [dagger] as defined above restricts to an antiautomorphism of [U.sub.P] for any symmetric poset. Furthermore,

U[O.sub.n]([F.sub.q]) [intersection] [U.sub.P] = {g [member of] [U.sub.P] | [g.sup.-1] = [g.sup.[dagger]]} and [uo.sub.n]([F.sub.q]) [intersection] [u.sub.P] = {x [member of] [u.sub.P] | -x = [x.sup.[dagger]]}.

Define [K.sub.u] and [[chi].sub.[lambda]] as in 1 and 2 with U = U[O.sub.n]([F.sub.q]) [intersection] [U.sub.P] and u = [uo.sub.n]([F.sub.q]) [intersection] [u.sub.P]. By Theorem 3.2, there is a supercharacter theory of U[O.sub.n]([F.sub.q]) [intersection] [U.sub.P] with superclasses {[K.sub.u]} and supercharacters {[[chi].sub.[lambda]]}. By Theorem 3.3, the superclasses are of the form [K.sub.g] [intersection] U[O.sub.n]([F.sub.q]) where [K.sub.g] is a superclass of Up in the algebra group supercharacter theory. In particular, if U is the unipotent radical of a parabolic subgroup of [O.sub.n]([F.sub.q]) then U = U[O.sub.n]([F.sub.q]) [intersection] [U.sub.P] for some symmetric poset P.

There are two important examples of a subgroup obtained from a symmetric poset in type D. First, let P be the symmetric poset on [2n] defined by

i [[??].sub.P] j if i [less than or equal to] j and (i, j) [not equal to] (n, n + 1).

Then U[O.sub.2n]([F.sub.q]) [intersection] [U.sub.P] = U[O.sub.2n]([F.sub.q]), and we get a second supercharacter theory of U[O.sub.2n]([F.sub.q]) 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 [e.sub.1,n] - [e.sub.n+1,2n] and ([e.sub.1,n] - [e.sub.n+1,2n]) + ([e.sub.1,n+1] - [e.sub.n,2n]) of u are in the same orbit under the action of [UT.sub.2n]([F.sub.q]) on u[o.sub.2n]([F.sub.q]), but in different orbits under the action of [U.sub.P] on u[o.sub.2n]([F.sub.q]).

We can also consider the poset P on [2n] defined by

i [[??].sub.P] j if i [less than or equal to] j [less than or equal to] n or n + 1 [less than or equal to] i [less than or equal to] j.

In this case, U[O.sub.2n]([F.sub.q]) [intersection] [U.sub.P] [congruent to] [UT.sub.n]([F.sub.q]), and the supercharacter theory obtained is the algebra group supercharacter theory.

4.2 Supercharacter theories of unipotent symplectic groups

Let J be the n x n matrix with ones on the antidiagonal and zeroes elsewhere, and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For q a power of an odd prime, define

[Sp.sub.2n]([F.sub.q]) = {g [member of] [GL.sub.2n]([F.sub.q]) | [q.sup.-1] = -[OMEGA][g.sup.t][OMEGA]}

along with the corresponding Lie algebra

[sp.sub.2n]([F.sub.q]) = {x [member of] [gl.sub.2n] ([F.sub.q]) | -x = -[OMEGA][x.sup.t][OMEGA]}.

Let [UT.sub.2n]([F.sub.q]) be the set of unipotent upper triangular matrices of [GL.sub.2n]([F.sub.q]), and [ut.sub.2n]([F.sub.q]) be the set of strictly upper triangular matrices of [gl.sub.2n]([F.sub.q]). Define

U[Sp.sub.2n]([F.sub.q]) = [UT.sub.2n]([F.sub.q]) [intersection] [Sp.sub.2n]([F.sub.q]) and u[sp.sub.2n]([F.sub.q]) = [ut.sub.2n]([F.sub.q]) [intersection] [sp.sub.2n]([F.sub.q]).

Define an antiautomorphism [dagger] of [ut.sub.2n]([F.sub.q]) by [x.sup.[dagger]] = -[OMEGA][x.sup.[dagger]][OMEGA]. Note that [dagger] satisfies the conditions required by Theorem 3.2, and furthermore

U[S.sub.p2n]([F.sub.q]) = {g [member of] [UT.sub.2n]([F.sub.q]) | [g.sup.-1] = [g.sup.[dagger]]} and u[sp.sub.2n]([F.sub.q]) = {x [member of] [ut.sub.2n]([F.sub.q]) | -x = [x.sup.[dagger]]}.

Define [K.sub.u] and [[chi].sub.[lambda]] as in 1 and 2 with U = U[Sp.sub.2n]([F.sub.q]) and u = u[sp.sub.2n]([F.sub.q]). By Theorem 3.2, there is a supercharacter theory of U[Sp.sub.2n]([F.sub.q]) with superclasses {[K.sub.u]} and supercharacters {[[chi].sub.[lambda]]}.

In [AN09a], Andre-Neto have also constructed supercharacter theories of U[Sp.sub.2n]([F.sub.q]). As was the case with the unipotent orthogonal groups, their supercharacter theories are the sets of the form U[Sp.sub.2n]([F.sub.q]) n Kg, where Kg is a superclass of [UT.sub.2n]([F.sub.q]) under the algebra group supercharacter theory. In particular, the following theorem follows from Theorem 3.3.

Theorem 4.2 The supercharacter theory of U[Sp.sub.2n]([F.sub.q]) defined above coincides with that of Andre-Neto in [AN09a].

We mention that we can also construct supercharacter theories of certain subgroups of U[Sp.sub.2n]([F.sub.q]) just as we did for U[O.sub.n]([F.sub.q]). We skip this construction as it is identical to the orthogonal case.

5 Supercharacter theories of the unipotent unitary groups

Let q be a power of an odd prime, and for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], define [bar.x] by [([bar.x]).sub.ij] = [([x.sub.ig]).sup.q]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the group of unitary n x n matrices over [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this section we construct a supercharacter theory of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] using Theorem 3.2 and calculate the values of the supercharacters on the superclasses.

5.1 Construction

The map [x.sup.[dagger]] = J[[bar.x].sup.t]J defines an antiautomorphism of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if we consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as an [F.sub.q]-algebra. This involution satisfies the conditions required by Theorem 3.2, and furthermore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Define [K.sub.u] and [[chi].sub.[lambda]] as in 1 and 2 with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By Theorem 3.2, there is a supercharacter theory of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with superclasses {Ku} and supercharacters {xA}.

As with the orthogonal and symplectic cases, by Theorem 3.3 the superclasses are of the form [K.sub.g] [intersection] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [K.sub.g] is a superclass of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] under the algebra group supercharacter theory.

We mention that we can also construct supercharacter theories of certain subgroups of U[U.sub.n]([F.sub.q]) just as we did for U[O.sub.n]([F.sub.q]). We skip this construction as it is identical to the orthogonal case.

5.2 Superclasses and supercharacters

In this section we describe the superclasses and supercharacters of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in terms of labeled set partitions. Recall that, for 1 [less than or equal to] i [less than or equal to] n, we define i = n + 1 - i. A twisted [F.sub.q]-set partition will refer to an [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-set partition [eta] of [[eta]] such that if i [??] j [member of] n then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In particular, if i [??] [bar.i] [member of] [eta], then a satisfies [a.sup.q] + a = 0. For more on labeled set partitions, see [Mar12].

Lemma 5.1 Each superclass of U contains exactly one element u with the property that f (u) has at most one nonzero entry in each row and column.

To each twisted [F.sub.q]-set partition n we assign the element [x.sub.[eta]] [member of] u defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

and the element [u.sub.[eta]] [member of] U such that f ([u.sub.[eta]]) = [x.sub.[eta]]. Note that [x.sub.[eta]] is in fact an element of u and has at most one entry in each nonzero row and column.

Corollary 5.2 The elements

{[u.sub.[eta]] | [eta] is a twisted [F.sub.q]-partition}

are a set of superclass representatives.

As there are equal numbers of superclasses and supercharacters, the supercharacters can also be parameterized by twisted [F.sub.q]-set partitions. Given a twisted [F.sub.q]-set partition, define An e u* by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 5.3 The set

{[[lambda].sub.n] | [eta] is a twisted [F.sub.q]-partition}

is a set of orbit representatives for the action of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [u.sup.*].

For a twisted [F.sub.q]-set partition, we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 5.4 The superclasses and supercharacters are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

5.3 Supercharacter values on superclasses

The goal of this section is to calculate [[chi].sup.[eta]]([u.sub.v]), where [eta] and v are twisted [F.sub.q]-set partitions. We will call a supercharacter elementary if it corresponds to a twisted [F.sub.q]-set partition of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with i [not equal to] [bar.j] or of the form [eta] = {i [??] [bar.i]} with [a.sup.q] + a = 0. In order to simplify calculations, we will show that every supercharacter can be written as a product of distinct elementary supercharacters. This is analogous to the method used in types A, B, C and D (see [And02, AN09a]).

For a twisted [F.sub.q]-set partition n, we can write [eta] as a disjoint union of twisted [F.sub.q]-set partitions of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with i [not equal to] [bar.j] or of the form {i [??] [bar.i]} with [a.sup.q] + a = 0. In other words, there exists m such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with each [[eta].sub.r] of the described form.

For two characters [chi] and [psi], define their product by ([[chi].sup.[psi]])(u) = [chi](u)[psi](u).

Lemma 5.5 With notation as above,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We now calculate the values of the supercharacters on the superclasses. First, we determine the dimensions of the elementary supercharacters.

Lemma 5.6 Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (with i [not equal to] [bar.j]) be a twisted [F.sub.q]-setpartition; then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [eta] = {i [??] [bar.i] (with i [less than or equal to] n + 1/2) be a twisted [F.sub.q]-setpartition; then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We mention that the dimension of an arbitrary supercharacter can be calculated by applying Lemma 5.5. Next we calculate the value of a supercharacter on a superclass.

Theorem 5.7 Let [eta] and v be twisted [F.sub.q]-set partitions. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [nst.sup.[eta].sub.v] = [absolute value of {i < j < k < l | j [??] [member of] v, i [??] l [member of] n}].

By Lemma 5.5, the proof reduces to proving that the theorem holds in the case that [[chi].sub.[eta]] is an elementary supercharacter. Note that in the above formula the sum of the terms ab is an element of [F.sub.q], even though each individual term might not be. This formula is identical to the type A supercharacter formula (which was first derived by Andre, and can be found in [Thi10]), except with -q replacing q everywhere (note that the supercharacter degrees are all powers of [q.sup.2]). This idea that information about the unitary group can be obtained from information about the general linear group by replacing q with -q is referred to as Ennola duality (see [Kaw85]).

References

[AN06] C. A. M. Andre and A. M. Neto. Super-characters of finite unipotent groups of types [B.sub.n], [C.sub.n] and [D.sub.n]. J. Algebra, 305(1):394-429, 2006.

[AN09a] C. A. M. Andre and A. M. Neto. A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups. J. Algebra, 322(4):1273-1294, 2009.

[AN09b] C. A. M. Andre and A. M. Neto. Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups. Pacific J. Math., 239(2):201-230, 2009.

[And95] C. A. M. Andre. Basic characters of the unitriangular group. J. Algebra, 175(1):287-319, 1995.

[And02] C. A. M. Andre. Basic characters of the unitriangular group (for arbitrary primes). Proc. Amer. Math. Soc., 130(7):1943-1954 (electronic), 2002.

[DI08] P. Diaconis and I. M. Isaacs. Supercharacters and superclasses for algebra groups. Trans. Amer. Math. Soc., 360(5):2359-2392, 2008.

[DT09] P. Diaconis and N. Thiem. Supercharacter formulas for pattern groups. Trans. Amer. Math. Soc., 361(7):3501-3533, 2009.

[[GKP.sup.+]90] P. M. Gudivok, Yu. V. Kapitonova, S. S. Polyak, V. P. Rud'ko, and A. I. Tsitkin. Classes of conjugate elements of a unitriangular group. Kibernetika (Kiev), (1):40-48, 133, 1990.

[Isa95] I. M. Isaacs. Characters of groups associated with finite algebras. J. Algebra, 177(3):708-730, 1995.

[Kaw85] N. Kawanaka. Generalized Gel'fand-Graev representations and Ennola duality. In Algebraic groups and related topics (Kyoto/Nagoya, 1983), volume 6 of Adv. Stud. Pure Math., pages 175-206. North-Holland, Amsterdam, 1985.

[Mar12] E. Marberg. Actions and identities on set partitions. Electron. J. Combin., 19(1):Paper 28, 31, 2012.

[SS70] T. A. Springer and R. Steinberg. Conjugacy classes. In Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics, Vol. 131, pages 167-266. Springer, Berlin, 1970.

[Thi10] N. Thiem. Branching rules in the ring of superclass functions of unipotent upper-triangular matrices. J. Algebraic Combin., 31(2):267-298, 2010.

Scott Andrews

University of Colorado Boulder, USA
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