Abstact. In this paper, we introduce a big family of finite groups , for which the orbit method works perfectly well. Namely, let be the algebraic group of upper unitriangular matrices with entries from , and be the finite field with elements. We define as the quotient of of the group over its second commutator subgroup.
To Leonid Makar-Limanov on the ocasion of his 80-th birthday
1 Introduction 
This paper is part of a bigger program of the application of the orbit method in representation theory. The main ingredient of the orbit method is the notion of a coadjoint orbit. The method works not only for ordinary Lie groups, but also for infinite-dimensional Lie groups and for algebraic groups over a topological ring , which satisfies the following condition. Every additive character of has the form , where and is a fixed non-trivial character.333In other words, the additive group is Pontryagin self-dual. This condition is satisfied for real, complex, quaternion, -adic and finite fields and also for the adele ./ring.
It is known (see. e.g. [1]) that the method can be formulated as a collection of simple rules, which gives transparent answers to all main questions. This βUserβs guideβ can be understood literally when the group in question is a connected and simply connected nilpotent Lie group. For many other Lie or algebraic groups it works after some algebraic and/or topological corrections. There are also groups, for which the right application of the orbit method is still unknown.
One time it was a hope that some modification of the orbit method will work for the groups of upper unitriangular matrices with elements from a finite field (considered as algebraic groups). A simple argument shows that the number of coadjoint orbits of the group is equal to the number of conjugacy classes, hence to the number of equivalence classes of irreducible representations of this group. However, the explicit construction of irreducible representations and their characters in terms of coadjoint orbits is known only for small .
In this paper, we consider a family of groups related to, but more simple than the full triangular group , for which all questions of representation theory have explicit answers in terms of coadjoint orbits. Namely, we consider the quotient group of by its second commutant . We call it βtwo-diagonal groupβ and denote by , or simply . It is a -dimensional affine algebraic group over the field . As such the group has the Lie algebra with the dual space . These spaces are the spaces of respectively, the adjoint and coadjoint representations of . Our first goal is to describe explicitly orbits of these two representations.
At the beginning, we impose no restrictions on the ground field , but out main results (Sections 2.2, 3.2, 4, 5) concern the case .
Both and can be realized as subquotients of the full matrix space Mat. We choose the coordinates for and for , so that
A general element of the group has the form
This justifies the name βtwo-diagonal groupβ). We denote this element by where and . The group laws are
For adjoint and coadjoint representations, the action of does not depend on . The diagonals of βs and βs are invariant with respect to the -actions, and the action on the diagonals of βs and βs is described by the formulas:
where the matrix and the matrix are
It will be more convenient to us to write the above formulas for the -action in a more traditional form:
![[Uncaptioned image]](f1.png)
We will refer to orbits of the coadjoint representation as just to βorbits,β and to orbits of the adjoint representation as to βclasses.β
2 Orbits. 2.1 Description. For given we consider the right system (1) as the system of equations with unknowns . If this system has a solution, then the matrices with and belong to the same orbit.
First consider the case, when are all different from zero. Our system splits into two systems with unknowns and unknowns , and this splitting looks differently in the cases of even and odd .
If n is even, then the two systems are
![[Uncaptioned image]](f2.png)
Both systems have unique solutions. We solve the first system βfrom the bottom to the topβ: we find from the last equation, then we find from the equation before the last, and so on, up to from the second equation and from the first equation. The second system can be solved βfrom the top to the bottomβ: we find from the first equation, then find from the second equation, and so on. Thus, in this case there is only one orbit (corresponding to the fixed set of non-zero elements of ). This orbit is an -dimensional vector space over with coordinates
If n is odd, then the two systems are
![[Uncaptioned image]](f3.png)
For the second system, the number of equations is one less than the number of unknowns. It has more than one solution: we can choose arbitrary and then find from our equations from the bottom to the top. However, for the first system, the number of equations is one more than the number of unknown. We can find using all the equations except the last one, , and the value of , which we have already found, may unfit this last equation. A simple computation shows that this equation is satisfied if the equality
![[Uncaptioned image]](f4.png)
holds. In other words, the expression in the left hand side of (2) is an invariant of the action of , and, in the case of odd , the orbit is characterized by the set of non-zero elements of and the value of our invariant (which may be 0).The dimension of the orbit in this case is with coordinates
Now let us turn to the general case: suppose that , while all the remaining βs are different from zero. Then the right system (1) splits into independent systems, :
![[Uncaptioned image]](f5.png)
![[Uncaptioned image]](f6.png)
Let us introduce additional notations: , and for . Thus, .
The systems have disjoint sets of unknowns: the unknowns in the system are (up to the numerations of βs, βs βs, and βs) are reduced copies of the right system (1), and we can apply to it our findings for that system. Thus, if is even, then the system has a unique solution , and if is odd then the system is consistent (although a solution is not unique) if and only if , where
and is defined by the same formula with all βs replaced by βs.
We arrive at the following description of all orbits of .
First, we need to fix an ordered partition
(where are positive integers). Then we put
Second, we choose such that and all the other βs are different from zero.
Third, we choose a for each from such that is odd.
These data determine an orbit. This orbit consists of the matrices with βs fixed above and arbitrary satisfying the condition for all such that is odd. If is the number of such then the dimension of this orbit is . Notice that and have the same parity, for an integer . Hence the dimension of the orbit is , in particular, the dimensions of all orbits are even.
If , then the number of orbits corresponding to an ordered partition is .
Notice that if is the number of even terms of our partition, then . Indeed, each even term of the partition is at least 2, and each odd term is at least 2. Hence .
2.2 The number of orbits of given dimension. We suppose again that . The following holds:
The number of orbits of of dimension is
The proof is based on the following combinatorial
The number of (ordered) partition of into the sum of even and odd positive integers is
Proof of Lemma 2.2. To specify a partition of into even and odd summands, we first need to choose positions for the even summands, which can be done in ways. Then we transform all our partitions in the following way: we add 1 to each odd summand and then divide all the summands by 2. We get an ordered partition of the number into summands with no condition on the parity of the summands. It is well known that the number of such partitions is , which implies our statement.
Proof of Theorem 2.1. According to Section 2.1, -dimensional orbits of correspond to (ordered) partitions of with the number of odd terms equal to . The number of orbits corresponding to partitions with even and odd parts is
By Lemma, the number of such partitions is
Thus the total number of orbits of dimension is
The sum in the last line is
We plug this expression for the sum into the last formula and see that the total number of orbits of dimension is
2.3 More on ordered partitions. Since ordered partitions of natural numbers played an essential role in the previous section, we present here a survey of their properties. Technically, the results of these section will be used only in Section 5 below, so the reader may postpone its reading until that section.
2.3.1. Partial ordering. Intervals. There is a partial ordering in the set of all ordered partitions of a number : a partition of precedes a partition (notations: or ), if the parts of are obtained from the parts of by a further subdivision. Notice that if is the number of odd terms of the partition and , then (there is no such inequality for even terms).
A more visualizable description of this partial ordering can be done in terms of dividers. Namely, an ordered partition of may be presented as a line of dots with dividers placed between some of them. For example, the diagram
![[Uncaptioned image]](f7.png){kind=small}
presents the partition of the number 11. In the language of such diagrams, the relation means that the diagram for is obtained from the diagram for by adding some (maybe, empty) set of additional dividers.
Notice also that if is a partition with terms, then the diagram for contains dividers.
If , then the interval , or the interval with the head and the tail , consists of all partition such that . In terms of the dots/dividers diagrams, the diagram for a in the interval contains all the dividers from the diagram for plus some (maybe, empty) subset of the set of the additional dividers of . If is the number of terms of the partition , then the number of these additional dividers is , and the set of additional dividers has subsets. Accordingly, the interval consists of partitions. For example, the set of all ordered partitions of is the interval with the head and the tail . So, the total number of ordered partitions of is .
Notice also that if the number of odd terms in the partition is , then the number of odd terms in any partition in the interval the inequalities hold. In particular, if and have the same number of odd terms, then every partition in the interval also has odd terms.
2.3.2. Partitions of even and odd types. The following definition looks artificial, but it will be crucially important in the last section of this article.
Definition. We say that the ordered partition of belongs to the even (odd) type, if the first , which is not equal to 1, is even (odd).
This definition becomes ambiguous for the partition , and we declare that this partition belongs to both even and all types. For example, if , then:
Partitions of the even type:
Partition of the odd type:
2.3.3. Numbers of ordered partitions Let be the numbers of ordered partitions of of, respectively, even and odd type. Since the full number of ordered partitions of is and one partition ( belongs to the both types, the sum must be . It is easy to find the numbers and for small values of :
![[Uncaptioned image]](f8.png)
To fill this table in, we can either to list and count all ordered partitions of either type (at it is done for above), or use the equality and the following simple
For every ,
Proof. To establish a bijection between the even type partitions of and the odd type partitions of , we put and assign to every other even type partition of the same partition of with the first even term increased by 1.
Using these equality and lemma, we fill the two middle rows of the table beginning with and following the path
![[Uncaptioned image]](f9.png){kind=small}
Here is the final result:
Proof. It is sufficient to check that , and . In all these cases the check is immediate.
In conclusion, we will provide two more calculations, which will be useful below (in Section 5). It will be the first, but not the last, example of calculations, which result in the Fibonacci numbers. For the Fibonacci numbers, we use the notation , where and , if .
We begin with two examples of partitions of even and odd types. First, a partition with all terms equal to 1 or 2 belongs to the even type. Second, a partition with all terms odd belongs to the odd type.
The number of ordered partitions of into 1βs and 2βs is .
Examples. For , partition: 1.
For , partitions: .
For , partitions: .
For , partitions: .
For , partitions: .
Proof of Proposition 2.6. Let the number of ordered partitions of into 1βs and 2βs be . Then, (see above). Furthermore, if a partition of into 1βs and 2βs ends with 1, then we remove this 1 and obtain a partition of into 1βs and 2βs; if it ends with 2, we remove this 2, and obtain a partition of into 1βs and 2βs. Moreover, any ordered partition of or of into 1βs and 2βs may be obtained in this way. Hence, , so .
The number of ordered partitions of into odd terms is .
Examples. For , partition: 1.
For , partition: .
For , partitions: .
For , partitions: .
For , partitions: .
For , partitions:
.
Proof of Proposition 2.7. Let the number of ordered partitions of into odd terms be . Then, (see above). Furthermore, if a partition of into odd terms ends with 1, then we remove this 1 and obtain a partition of into odd terms; if the last term is 3 or more, we subtract 2 from this term, and obtain a partition of into odd terms. Moreover, any ordered partition of or of into odd terms may be obtained in this way. Hence, , so .
2.4 Basic orbits. We will call the orbits corresponding to the sets of non-zero βs basic. Then, in some sense, all orbits are products of basic orbits. Let and be as in Section 2.1. Consider the projection
presented graphically (in the case ) in the diagram below:
![[Uncaptioned image]](f10.png)
The kernel of the projection (3) is the central subgroup of described by the conditions: all βs and βs are zero, except ( and in the diagram above).
Thus the entries of the group from this construction () acquire the notations ; its adjoint representation is defined in the space of upper triangular matrices with the entries ; and its coadjoint representation is defined in the space of lower triangular matrices with the entries .
For fixed non-zero , plus one additional invariant if is odd (see Section 2.1), there arises a basic orbit of the group ; fix such orbit for every . The product of these orbits lies the space of the coadjoint representation of the product and is an orbit of this product group. The projection (3) gives rise to the action of the group in this orbit, and with this action, it is not different from the orbit of corresponding to the partition and the set of βs,
![[Uncaptioned image]](f11.png)
3 Classes 3.1 Description The values of are fixed within a class, so we need to consider possible equivalences between the sequences for every fixed sequence . The sequences and are equivalent, if the left system (1),
has a solution in . If all βs are non-zero, then this system has a solution, so in this case there is one class of capacity and the set of these classes is labeled by elements of . Let us look, what happens if some are zeroes.
If , but and (in particular, ), then the equations with and become
which implies
Thus for the consistency of our system it is necessary that the last equality holds. In other words, is an invariant, that is, it is fixed within a class.
We say that the set forms a string of zeroes of length if the following holds: ; if then ; if then . If is string of zeroes of length , then our system contains a part
(the first line is absent if , and the last line is absent, if ). Thus for the consistency of our system it is necessary that . In other words, are invariants.
Below, we will refer to non-zero as to -invariants, and to invariants involving as to -invariants.
Thus, for any fixed we have a full description of invariants; this description depends only on the locations of zero βs. For example, if and , while , then, in the addition to -invariants , there are 4 -invariants: , and a class is determined by fixing their values. Hence in this case classes are parallel 5-dimensional affine planes in the 9-dimensional space spanned by .
3.2 The number of classes of given dimension. In Section 2.2, we proved a compact formula for the number of -orbits of a given dimension. Unfortunately, no formula of this quality exists for the classes. Below, we restrict ourselves to an inductive procedure of calculating the number of classes of a given dimension. We again assume that . An element of is characterized by elements of . The string is fixed within any class. We will usually label such a string by a sequence of heavy dots and light dots : heavy dots correspond to non-zero and light dots correspond to zero . We will denote the number of -dimensional -classes by . More specifically, and will denote the number of -dimensional -classes with, respectively, and . (Thus, ).
For a given , there are strings of heavy and light dots. For each such string we denote the number of heavy dots by and the number of invariants, calculated as in Section 3.1, by . Then this string contributes classes into or , if, respectively, the first dot is heavy or light. For example, the string
contributes classes into .
How to find or, separately, and ? There is a βdirectβ way of doing that: to consider all strings of heavy and light dots, repeat for each of them the computations similar to those in (4) and calculate the appropriate sums. Let us do this, for example, for :
![[Uncaptioned image]](f12.png)
From this:
However, calculation like this for large is hardly possible. Below, we deduce an expression of in terms of and , which can be used for calculating step by step.
For ,
ProofΒ Let us look how , and in (4) change, when we cut off one or two first dots from the string of heavy and light dots.
![[Uncaptioned image]](f13.png)
Since every string of heavy and light dots with begins with one of the combinations in the first column of the table above, the statement of Theorem follows.
The relations in Theorem 3.1 determine the induction step. The base of induction is provided by computations for given above (and still easier computations for and 2).
3.3 M-classes. This section, like Section 2.3 above, will not be used before Section 5.
3.3.1. Sparse sequences and the definition of M-classes. A subset of is called a sparse sequence, if for every . Let us count the number of sparse sequences.
For a given , the number of sparse sequences is the Fibonacci number .
Examples. For , there are sparse sequences: .
For , there are sparse sequences: .
For , there are sparse sequences: .
For , there are sparse sequences: .
Proof. For see above. For , let us consider a sparse sequence in . If the last term of this sequence is not , then it is also a sparse sequence in . If the last term is , then the sequence, being sparse, does not contain . So, if we remove the term , we obtain a sparse sequence in . Thus the number of sparse sequences in is the sum of the numbers of sparse sequences in and sparse sequences in .
Definition. A class is called an -class, if the subscripts of -invariants form a sparse sequence.
3.3.2. Containers For a sparse sequence , we denote by the subset of , which consists of all matrices with . Thus, is a union of -classes. Containers play for -classes a role similar to the role which ordered partitions play for orbits: they place together -classes with some important properties being the same. First of all, the -classes from the same container have the same stabilizer.
The formulas for the multiplication and inversion in the group imply the following formula for the conjugation:
Hence belongs to the stabilizer of if and only if
which means precisely that, for every , (here we use the fact that is sparse and mean that ). This shows that all elements of have the same stabilizer, in other words, all -classes within have the same stabilizer, and this stabilizer is a normal subgroup of . For a more convenient description of , we need new notation. Let be the set of those , for which or is contained in , and let be the complement of in . Then is the set of all with , if , that is, may be different from zero only if . (Notice that may be not sparse and that .)
The group may be non-commutative. Its commutator subgroup consists of all with all βs and βs being zero, with a possible exception of those , for which .
Our description of the container , classes , and the stabilizer give rise to direct computations of cardinalities of several related sets.
(i) The number of containers is .
(ii) .
(iii) For a class ,
(iv) For a class , the number of -invariants is .
(v) The number of classes in is .
Proof. Part (i) is the same as Proposition 3.2.
Part (ii) follows from our description of : an element of this group is determined by .
Part (iii) follows from Part (ii), since for
can be calculated also directly form the description of : the entries are fixed within , and the entries belong to a subspace of the -space, whose codimension is the number of -invariants. Hence, . Compare the two computations of : ., as stated in Part (iii).
Part (iv) follows from Part(iii): a class in is determined by (non-zero) -invariants and -invariants (which are allowed to be zero).
4 Representations. The orbit method, as it is described in [1], provides a bijective correspondence between coadjoint orbits and unireps. Our goal is to show that it works for the group .
We begin with the observation that the construction of a complex representation corresponding to a given orbit works for this group. We will provide an explicit construction of a representation of corresponding to an orbit and an explicit calculation of the character of this representation.
Then we will show that all the representations constructed are irreducible, and the set of these irreducible representations is complete in the sense that any irreducible complex representation of is equivalent to precisely one of the representations constructed.
Our construction will involve a fixed non-trivial homomorphism of the additive group of the field into the multiplicative group . Notice that the image of this homomorphism is contained in the set of degree roots of 1. Notice also that this homomorphism is not unique: there are such homomorphisms labeled by non-zero elements of : the homomorphism acts by the formula
4.1 The construction of representations by the orbit method. The dimension of a representation corresponding to a -dimensional orbit is . The construction consists of three steps. Let be an orbit.
First step. We choose a representative , compute the stabilizer , and choose a polarization subgroup corresponding to some such that
The last property shows that the formula defines a 1-dimensional unitary representation of the group .
Second step. Consider the right homogeneous -space and the natural projection If we choose for every class a representative , then every element can be uniquely written in the form where . Clearly, here is the class and . Thus the function allows to identify with the direct product : an element corresponds to the pair .
Third step. Introduce the so-called Master Equation:
We have seen above that this equation has a unique solution, .
Now we can describe the representation of . The space of this representation is the space of complex-valued functions on ; since has dimension over , the (complex) dimension of the space of function is . The representation is described by the formula
4.2 Basic representations and their characters. We begin with the representations of corresponding (in the sense of Section 4.1) to the basic orbits (Section 2.4). We begin with the (simpler) case of even .
4.2.1. The case of even n. In this case, basic orbits are labeled by sets of non-zero elements of the field . Values of within every orbit are arbitrary, and for a representative of the orbit corresponding to we take the matrix
![[Uncaptioned image]](f14.png)
(Thus the stabilizer of does not depend on .)
For , we can take the group consisting of matrices shown below (the condition holds, because the group is commutative). The space has dimension ; the elements of this space are represented by matrices also shown below.
![[Uncaptioned image]](f15.png)
The Master equation becomes
![[Uncaptioned image]](f19.png)
(the products of matrices are βtruncatedβ: we annihilate all the entries above the -diagonal). We equate the entries of the two product matrices and obtain the equalities:
Now we have a description of the representation of corresponding to the basic orbit corresponding, in turn, to the set of non-zero elements of . The space of this representation is the -dimensional space of complex-valued functions , and the representation is defined by the formula
Let us calculate the character of this representation. The space has a natural basis composed of β-functionsβ: for a fixed set , the -function takes the value 1 on this set of variables and the value 0 on other sets of variables. The entries of the matrix of the operator with respect to this basis correspond to pairs , and this entry is zero, if . Thus, if at least one of is not zero, the matrix of has no non-zero diagonal entries, and the trace of this matrix, that is, the value of the character, is zero.
Suppose that . Then the diagonal entry of the matrix corresponding to is
The sum of all these diagonal entries is
Finally, notice that for a ,
But only if . We arrive at the final result: the value of the character of the representation on the matrix
![[Uncaptioned image]](f16.png)
is if , and is zero otherwise.
4.2.2. The case of odd n. In this case, a basic orbit is determined by a set of non-zero elements of the field and a value of the invariant
For a representative of we can take the matrix
![[Uncaptioned image]](f17.png)
The stabilizer of consists of matrices
![[Uncaptioned image]](f18.png)
with and satisfying the system of equations . In other words,
for some . For , we can take the group of matrices shown on the left below (no relations between ; again, the condition holds, because the group is commutative). The space has dimension and the representatives of elements of are shown below on the right.
![[Uncaptioned image]](f20.png)
The Master equation has the same appearance as before (see (5)), but the matrices now have even, not odd, order. For the entries, we have the following equalities:
Now we have a description of the representation of corresponding to the basic orbit corresponding, in turn, to the set of non-zero elements of and the value of the invariant (6). The space of this representation is the -dimensional space of complex-valued functions , and the representation is defined by the formula
Let us calculate the character of this representation. The space again has a basis of -functions. The entries of the matrix of the operator with respect to this basis correspond to pairs , and this entry is zero, if . Thus, if at least one of is not zero, the matrix of has no no-zero diagonal entries, and the trace of this matrix, that is, the value of the character, is zero.
Suppose that . Then the diagonal entry of the matrix corresponding to is
First of all,
The sum of all these diagonal entries is
Each of the sums in the last formula equals , if , that is, if the equalities (7) hold, and at least one of them is zero otherwise.
It remains to find in the case, when (7) holds. But in this case
We arrive at the final result: the value of the character of the representation on the matrix is if and the formulas (7) hold for . In all other cases, the value of the character is zero.
4.3 Representations corresponding to all orbits. We observed in Section 2.4 that all orbits are products of basic orbits. More precisely: if an orbit of corresponds to an ordered partition , then this orbit is the products of some basic orbits of the groups with the action of determined by the projection (3). But in this case the representation of determined by these orbits is the tensor product of the basic representations of the groups corresponding to the orbits . And the character of the representation is the product of characters of the representations , which were calculated in Section 4.2.
Let us prove now that the representations constructed are all irreducible, and any irreducible representation is equivalent to one of them.
Since all these representations have different characters, they are not equivalent to each other. Let us prove now that
4.3.1. They are all irreducible. Let us begin with basic representations (Section 4.2). The space of each of these representation is the space of complex-valued functions if is even and , if is odd; the variables in both cases are elements of the field . For a basis of this space we can take the set of β-functions;β each of them takes value 1 for some fixed set of variable and the value 0 on every different set of variables. It is clear that if is a -function concentrated at , and , then, is the -function concentrated at times a non-zero coefficient. This means that if a space of a subrepresentation of our representation contains a -function, then this space contains a whole basis, that is, this subrepresentation coincides with the whole representation.
What if the space of a subrepresentation contains no -functions? Take a non-zero element of this space, which is a linear combination of the minimal possible number of -functions (that is, takes non-zero values at points). We can apply to it an operator , where is a matrix with , Then the result will be a linear combination of the same -function as (that is, the function takes non-zero values at the same points as ). More precisely, the value at is multiplied by
(, since ) times , if is even.
Suppose that takes non-zero values at the points . If , then we apply to this the matrix with . The values will be multiplied by different numbers. If , but , then we apply with and all other βs are zeroes β again will be multiplied by different numbers. And so on. Thus and . Let . The function is not zero, belongs to the space of the subrepresentation, and takes non-zero values at less than points (since ). This contradicts the minimality of . Thus, the space of the subrepresentation contains a -function, hence the subrepresentation coincides with the whole representation, hence this whole representation is irreducible. This completes the proof of irreducibility of all basic representations.
The construction of a general representation is described in Section 2.4. We fix an ordered partition
of and consider the projection (3)
(see Section 2.4). Then we fix basic orbits of the groups and corresponding representations of these groups. The space of the representation is the space of complex-valued functions of variables from . The tensor product
becomes the space of representation of , and the projection (3) turns it into a representation of ; this is a construction of a general representation of . Let us prove that this representation is irreducible.
The basis of the space consists of -functions. Hence the basis of the space consists of tensor productswhere is a delta-function from . Take a non-zero subrepresentation of this representation. We want to prove that the space of this subrepresentation contains a vector from the basis. Take a non-zero vector of this space. It is a linear combination of the vectors of the basis. Suppose that this linear combination involves more than 1 basic vectors, say, it involves different and (where all are -functions). Then for some . We have seen before that there is some , which takes every -function into itself times a non-zero coefficient and, in particular, . Let . Then is a linear combination of the same basic vectors as except , thus it involves one less basic vectors than . Repeating this operation, we arrive at a basic vector, which is contained in . Appropriate elements of take this basic vector into all other basic vectors. Thus, , and our representation is irreducible.
4.3.2. The list of irreducible representation is complete To prove this, we need to check that the sum of the squares of dimension or the representations constructed is equal to the number of elements of the group , that is, to . The dimension of the representation corresponding to a -dimensional orbit is , the number of -dimensional orbits was calculated in Section 2.2 (Theorem 2.1). Thus, we need to prove the following equality:
or
Put
(In particular, ). Then the left hand side of the equality (8) is the sum of and
On the other hand,
From this:
as was stated.
5 Models 5.1 Introduction For a finite group , a model is a finite dimensional representation, whose decomposition into irreducible representations contains an irreducible representation of every equivalence class precisely once. The formal definition of a model (as well as the term model) was first introduced by I.N.Bernstein, I.M. Gelfand and S. I. Gelfand in their article [2]; its significance for the representation theory was demonstrated later by I.M. Gelfand and A.V. Zelevinsky [3].
If we have a classification of irreducible representations of (which is the case for ), then there is a ready construction of a model. What we are interested in here is a βgeometricβ construction of a model for the group , the meaning of which we describe below. This description does not use any specific properties of the group , so we are speaking of an arbitrary group .
We fix an ad-invariant (that is, consisting of whole classes) set and consider the union of 1-dimensional complex vector spaces . Then we fix a lifting of the adjoint action of in to the action in : for a , the transformation maps isomorphically onto . The space of the representation of , which is the goal of our construction is the space of βtwisted functionsβ . The action of in is defined by formula Our goal is to choose an and the lifted action of in in such a way that the representation of in be a model for .
Return to the general case. There are two ways of understanding construction above: algebraic and geometric.
The algebraic way, which we, actually, follow below, is the following. For every conjugacy class , choose a representative and consider the stabilizer of . Then for maps into , forming a 1-dimensional representation of . For , the map is the multiplication by a non-zero complex number , and the homomorphism is the character of the representation . These determine (up to an isomorphism) the whole action of in , and the representation becomes the direct sum of induced representations.
The geometric way fits better the topological case. The union may be regarded as a line bundle over , and the action of in is a fiberwise lifting of the action of in . Twisted functions are sections of the bundle, and the action is the natural action.
This way of constructing a model was used by A. A. Klyachko [4] in his construction of a model for the symmetric group . For , Klyachko used the set of all βinvolutionsβ . (Later the significance of the manifold of involutions in the representation theory was demonstrated by Anne Melnikov [5].) In our construction of a model for the group , we follow this idea: is precisely the set of involutions in the case . For , there are no involutions in (besides the identity), and we need to modify the definition of . We do it in the section 5.2 below.
5.2 The case of : the set M, stabilizers and characters. For the set (see Section 5.1) we take the set of matrices with for . Thus, the subscripts , for which must form a sparse sequence in . In other words, is the union of all -classes, which we studied in details in Section 3.3. In Section 3.3.2, we formed unions of -classes, which we called containers. Containers correspond to sparse sequences ; the container consists of those , for which if and only if . Every -class belongs to one container.The number of containers in is the Fibonacci number .
Recall two useful definitions from Section 3.3. We denote by the set of those , for which either or belongs to . It is obvious that The complement is denoted by . A class is determined by -invariants and -invariants, which are allowed to be 0 (see Sections 3.1 and 3.3.2). Thus, the container contains classes; every class in contains elements of .
All have the same stabilizer, which we denote by . We repeat the description of from Section 3.3.2. Let be the set of those , for which either or belongs to . It is obvious that The complement is denoted by . The stabilizer consists of such that for , that is, may be different from zero only for .
The group is normal, but may be non-commutative. Its commutator subgroup consists of all with all βs and βs being zero, with a possible exception of those , for which . Hence, every homomorphism has the form
where , for , and , if . (Recall that is a fixed non-trivial homomorphism of the additive group of the field into the multiplicative group β see Section 4.) Thus, the construction of a model will involve a choice of the coefficients and for every -class. This choice will be done in Section 5.X.
In conclusion, we provide a formula for the character of the representation of induced by a 1-dimensional representation of with the character (9).
According to the classical Frobenius formula, if is the character of some finite-dimensional representation of a subgroup of a finite group , then the character of the induced representation of is described by
If the subgroup is normal (which is our case), then the description becomes
The formulas for the group operations in (Section 1) imply the formula for the conjugation:
In our case, the Frobenius formula gives (we use the notations and :
The expression under does not depend on βs and on with , so the formula may be simplified:
Since ,we arrive at the final result:
5.3 Flocks. 5.3.1. Introduction. The construction of a model for the group , which we outlined in Section 5.1, is supposed to make every irreducible representation of an irreducible component of a certain representation of induced from a 1-dimensional representation of the stabilizer of a certain M-class . This would determine a mapping
![[Uncaptioned image]](f21.png)
This map, actually, will take irreducible representations related to one ordered partition of (see Section 4.3) into -classes from one container (we will see in Section 5.X that there will be one small exception to this rule), the mapping (10) will be a refinement of a certain mapping
![[Uncaptioned image]](f22.png)
For every , the number of containers is the Fibonacci number , while the number of partitions is . If is large enough (), then (for example, for these two numbers are 32 and 21, and for they are 512 and 144). So, we must be prepared to the fact that several partitions will be assigned to the same container. For the sets of partitions, which are going to be assigned to one container, we will use the term flocks. We will begin with a descriptions of flocks.
5.3.2. Description of flocks. In Section 2.3.2, we introduced a definition of two types of (ordered) partitions: even and odd. Remind that an ordered partition belongs to the even (odd) type, if the first (if such exists) is even (odd). According to this definition, the partition belongs to the both types.
Remind some useful notations from Section 2.3. The numbers of even and odd terms in the partition of are denoted, respectively, by and ; the numbers and are of the same parity, and number is denoted by .
All the partitions within a flock will be of the same, even or odd, type. Accordingly, we will speak of flocks of even or odd type. Every flock will be an interval with respect to the partial ordering of partitions (see Section 2.3.1), that is, it has a head and a tail and consists of all partitions such that .
The head of a partition of odd type must be a partition into odd parts and no even parts. The tail of a partition of odd type must have the first term greater than 1 (if there is any) equal to 3, and all the other terms equal to 1 or 2. To obtain the tail from the head we replace the first by and every other by . For example, if the head is , then the tail will be . Every partition of the odd type belongs to precisely one flock of the odd type. To obtain the head of this flock we need to combine every odd with all even terms after it (and before the next odd term); to obtain the tail of this partition, we need to replace the first odd term greater than 1 by , every other odd term by and every even term by .
The head of a partition of even type must begin with even term and have all the terms after that odd. The tail of a partition of the even type must consist of 1βs and 2βs. To obtain the tail from the head, we replace the even term (if there is any) by and replace every odd term by . For example, if the head is , then the tail will be . Every partition of the even type belongs to precisely one flock of the even type. To obtain the tail of this flock, we need to replace every odd term by and every even term by . To obtain the head of this flock we keep the 1βs before the first term greater than 1 unchanged, then combine the even terms after these 1βs and before the next odd term, and then combine every odd term with all the even terms after it (and before the next odd term. For example, the partition of the even type belongs to the flock of the even type with the head and the tail .
Notice that the number of odd terms (but not the number of even terms!) and hence the number are fixed within the flock (of any type).
Notice also that it is convenient to count flocks of the odd type by heads and flocks of the even type by tails. Namely, the number of flocks of the odd type is the number of partitions of into odd parts, which is (Proposition 2.7). And the number of flocks on the even type is the number of partitions of into 1βs and 2βs, which is (Proposition 2.6). Thus the whole number of flocks is , which conveniently coincides with the number of -classes (see Section 3.3.1).
5.3.3. Pictures and examples It is convenient to present this procedure on a picture.
First we draw heavy dots in line. Then we put dividers between some dots, so the line is divided into parts, and this partition is the head of a flock. To obtain the tail, we should add some dividers, and we draw them as dotted lines. The description of flocks given above provides an instruction for choosing places for dotted dividers. Namely, for flocks of the even type, we divide every even part into and every odd part into . For the flocks of the odd type, the rules are the same with one exception: we divide the leftmost (odd) part of length into . It easy to prove (we leave this to the reader) that the number of dotted lines will be . All the partitions in our flock are obtained by using all solid dividers and some subset of the set of dotted dividers. Since the set of dotted dividers have subset, our flock will contain partitions.
Two examples of this construction are shown below (, odd type, and in the picture on the left, and , even type, and in the picture on the right.
![[Uncaptioned image]](f23.png)
Below, we show this splitting of the set of partition into flocks for and .
5.3.4. The cases n = 6 and 7 n = 6, odd type
![[Uncaptioned image]](f24.png)
n = 6, even type
![[Uncaptioned image]](f25.png)
n = 7, odd type
![[Uncaptioned image]](f26.png)
n = 7, even type
![[Uncaptioned image]](f27.png)
Notice that the partition belongs to both even and odd types and is considered as two flocks. Notice also that the numbers of partitions of the odd and even type (8 and 13 for and 13 and 21 for ) agree with the computations in the end of Section 5.3.2.
In the diagrams above, each flock is accompanied by the indication of the container, which is assigned to this flock; we will explain a way to determine this container in the next section.
5.3.5. The correspondence containers flocks. For every flock, we need to assign a container with some sparse sequence . We do this in the following way.
First of all, to flocks of the odd (even) type, we will assign a sparse sequence, which contains 1 (which does not contain 1).
If the flock is of even type, then its closing partitions consists of 1βs and 2βs. More precisely, there are 1βs and 2βs. Let be the numbers of parts equal to 2 (we assume that ). Then the container corresponding to this flock is . (Notice that the total number of parts is , so .)
If the flock is of odd type, then in its closing partition the first part greater than 1 is 3, and all the subsequent parts are 1βs and 2βs. Thus, in this partition parts are 1βs, parts are 2βs, and one part is 3. Let be the number of the part 3, and are numbers of parts equal to 2 (again we assume that ). The container corresponding to this flock will be .(Notice that the total number of parts is -1, so .)
(We should remark that the partition appears as both a flock of the odd type and a flock of the even type. As such, it is assigned to the container and to the container .)
It is easy to check that the choice of containers in the cases and shown in the diagrams in Section 5.3 agrees with the rules described here.
5.4 Characters of the representations in a flock. The main idea is that for partitions from the same flock, the characters of corresponding irreducible representations look almost the same. We will explain this on the examples considered in Section 5.3.2.
5.4.1. The case of the even type. We begin with the example of the even type flock with the head and the tail from Section 5.3.3. Let us supplement the dots/dividers diagram corresponding to this flock with the notations (corresponding to the dots) and (corresponding to the spaces between the dots):
![[Uncaptioned image]](f28.png)
The characters of representations corresponding to the partitions from this flock were computed in Section 4.2. The formula begins with the factor , contain the factors for first at every odd term in the head partition ( and in our example, overlined in the diagram and for all βs except those corresponding to the solid dividers (marked with crosses in the diagram). Also, in the diagram we underlined the βs, which correspond to the dotted dividers; they may disappear in the characters of representations, corresponding to non-head partitions in the flocks. It is convenient to assume that the underlined βs are allowed to be zero, while the terms not underlined must be different from zero. So, here is the character:
if and , and is zero otherwise. It is worth noticing that we underline every second term with .
The transition from the example to the general case is straightforward. We will only add the formulas for the numbers of the terms with βs and underlined and not underlined terms with βs.
The number of overlined β s (same as the number of factors with βs in the formula) is the number of odd terms in the partitions of the flock: it is .
The number of crossed βs is the number of solid dividers, which is one less that the number of terms in the head partition; so, it is also . Thus the number of βs in the formula for the character is .
The number of underlined -factors is the number of dotted dividers is (see Section 5.3.3). Hence, the number of -factors not underlined is .
5.4.2. The case of the odd type. Again, we begin with the example of the odd type flock from Section 5.3.3. The head and the tail of this flock are and (thus, ). We consider the dots/dividers diagram similar to that in Section 5.4.1.
![[Uncaptioned image]](f29.png)
The formula for the character is obtain according to he same rules as in the even case:
if and , and is zero otherwise.
The number of overlined βs is the same as in the even case: (it is the number of odd terms in the partition).
The number of crossed βs is again the number of solid dividers, thus it it one less than the number of terms in the partition. But now it is , so the number of crossed βs is . Thus the number of βs in the formula for the characters is .
The number of underlined -factors is again the number of dotted dividers is . Hence, the number of -factors not underlined is .
5.5 The end of the construction of a model. It remains to specify the one-dimensional representations (the characters) of the stabilizer of the classes within the container corresponding to a flock. Again, we will give the details of the construction for the examples of Section 5.3.3 and then discuss the general case.
5.5.1. The case of the even type; the example. Again, we begin with the example of the flock with the head and the tail from Section 5.3.3. According to Section 5.3.5, the container corresponding to this section is , where . In the notations of Section 3.3.2, and . The classes in this container are characterized by
4 -invariants, and
4 -invariants, .
The characters of representations induced by 1-dimensional representations of the common stabilizer
of classes in are
if for and and 0 otherwise.
The characters of representations corresponding to our flock are
where βs and βs are element of , different from 0 for βs not underlined. We need to choose values for βs and βs in terms of - and -invariants to accommodate the formulas (10) and (11). For values of we take (in any order) the -invariants . For , we take (in any order) three of the four -invariants, say, . The remaining -invariant, , we take for the value of (taking into consideration the fact that appears in both formulas (10) and (11)). For the remaining βs and βs in (10) we assume zero values.
Still two discrepancies in the formulas (10) and (11) remain. First, the first factors in (10) and (11) are different: and . Second, (11) contains terms with and , while (10) does not. Both can be eliminated by the following move. We remove the conditions and replace by Then the character (10) (of capacity ) becomes the sum of characters (11) of -dimensional representations labelled by .
5.5.2. The case of the odd type; the example. Now we consider the flock with the with the head and the tail (see Section 5.3.3). According to Section 5.3.5, the container corresponding to this section is , where . In the notations of Section 3.3.2, and . The classes in this container are characterized by
5 -invariants, and
5 -invariants, .
The formulas similar to (10) and (11) are
and
The coefficients in the non-underlined factors in (13) must be non-zero, so we used for them the -invariants: for . For , we use the -invariants, say, , and the remaining -invariants we assign to the coefficients in the terms with and , which appear in the both formulas (12) and (13). All the other coefficients and in (12) are assumed to be zero.
The last step repeats the last step in the previous case: we remove the conditions and replace in (12) by
5.5.3. General case Let us consider in the general case the construction of the character of the stabilizer which was described above for examples.
First we take the non-underlined factors in the formulas like (11) and (13) and assign for their coeficients the -invariants. But the numbers of non-underlined terms and -invariants agree: they both are for the even type and for the odd type. Thus, this step works in the general case.
Next we use -invariants for the underlined factors . The number of these factors is in the even case and in the odd case, while the total number of -invariants is always . But , so , hence there are sufficient quantity of -invariants. Moreover, -invariants in the even case and -invariants in the odd case and remain for the next step.
Next step consists in assigning the remaining -invariants to the coefficients to , which appear both characters we consider. We need to check that the numbers of these -invariants and these are the same. The cases of the even and odd types are slightly different. In the even type case the number of remaining -invariants is . The elements of correspond to those 1βs ion the tail partition, which do not follow 2; the same 1βs correspond to the common in the two characters. The only difference in the odd type case is that 1 belongs to , and hence not to , while appears in the both characters. This leads to the additional 1 in the expression for the number of common βs in the two characters.
And the last step: the difference between and in the two characters should be compensated by the unused part of the number of in the character of the representations in the flock . In is easy:
5.5.4. The possible non-commutativity of the stabilizer does not matter. If the stabilizer is not commutative, then some -factors in the formula for the character of the stabilizer (like (10) or (12)) may be missing. But this does not affect our construction. The reason is that this missing βs appear, when there are gaps in of the length 3 or more (even 4 or more, if this gap inside I, not in one of the ends). In this case, there are at least two 1βs in a row in the tail partition of the flock, they have to be separated by a solid divider (there are at most one 1 between any two successive dividers), and the corresponding is missing also in the character of the representation in the flock (formula like (11) or (13)).
References
