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Abstract
We determine character tables for twisted fractional linear groups that form the “other” family in Zassenhaus’ classification of finite sharply 3-transitive groups.
1. Introduction and preliminaries
A classification of all finite, sharply 3-transitive permutation groups follows from a classical result by Zassenhaus [Citation17], by which the only two families of such groups are the fractional linear groups for any prime power q and their “twisted” companions
for any odd prime power q. Recall that
can be introduced as the group of fractional transformations
of the set
where
with the obvious rules for calculations with
In the case of a finite field F of the form
for an odd prime power q one may “twist” the fractional transformations by considering the permutations of
defined by
if
and
if
where S(F) and N(F) are the sets of non-zero squares and non-squares of F and σ is the unique involutory (Galois) automorphism of F. The collection of all such “untwisted” and “twisted” fractional transformations under composition constitutes the twisted fractional linear group
The notation comes from the monographs [Citation12, p. 261] and [Citation6, p. 163], and is also used in the textbooks [Citation13, p. 188] and [Citation14, p. 283]; the letter M was introduced in the original article [Citation17, p. 36] as a tribute to Mathieu who discovered the first group in this series (for q = 3, of degree 10). A possible alternative notation could be derived from the existence of just three non-trivial 2-extensions of
by outer automorphisms—a diagonal automorphism δ, the Galois automorphism σ, and their product
—leading to the groups
and
see [Citation16].
While the fractional linear groups and their subgroups
have been studied widely, their twisted versions
have received comparatively less attention, and this also applies to characters. A number of resources on representations and characters of fractional linear groups are accessible (either in form of explicit tables or in terms of methods of their derivation, cf. [Citation2, Citation9]). Character tables for the “other” family of sharply 3-transitive groups, by contrast, do not appear to be available.
Among numerous applications of characters we mention here the ones related to the theory of regular hypermaps on compact surfaces, or, equivalently, finite groups generated by three involutions. In [Citation1], enumeration of regular hypermaps of a given type on fractional linear groups was obtained with the help of the Frobenius’ character formula [Citation4] for counting tuples of group elements with entries in given conjugacy classes; more applications of this type can be found in [Citation8]. There is, however, no corresponding result for regular hypermaps on twisted fractional linear groups, although enumeration of regular maps (of unspecified type) on these groups can be found in [Citation3].
The purpose of this paper is to calculate character tables for the twisted fractional linear groups for any odd prime power q. This will be done in a completely elementary way by means of standard results in representation theory, by inducing characters from subgroups of
and finding their decomposition into irreducible constituents.
To avoid fractional transformations we will work with a representation of used in [Citation3]), which also differs from the one of [Citation16] and which will prove more suitable for our purposes. For
q an odd prime power, we let
with
identified with the additive group C2 in the obvious way and with multiplication given by
where
is obtained from B by applying σ to every entry. For each
let
be defined by
if
and
if
The “twisted” subgroup K of J of index 2 is defined by letting
with multiplication as before, that is,
for every
Consider now the subgroup of K of index 2. The center L of K0 consists of pairs
where
is a scalar matrix; note that L is also normal in both K and J. The factor group
turns out to be isomorphic to
We will identify G with
throughout. This way, G can be regarded as a subgroup of index 2 of the group
and the factor group
can be identified with
Elements
that is, cosets
of the factor group
will throughout be denoted
they will be called untwisted if
and twisted if
2. Conjugacy classes of ![](//:0)
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To be in position to consider characters of the twisted group we will need to determine conjugacy classes of G. This was done in [Citation3] for conjugacy of twisted elements with respect to
and it turns out that the detailed analysis therein furnishes all one needs to determine the conjugacy within G, a subgroup of
of index two. We sum up the corresponding results in what follows, using notation and machinery of [Citation3]. For any
we let
and
be the 2 × 2 matrix over F with, respectively, the diagonal and off-diagonal entries a, b and with remaining entries equal to zero.
We begin by conjugacy of untwisted elements of forming the subgroup
of G. Every untwisted element of G can be identified with
for
or, for short, just with A, tacitly assuming that writing
means
for
as usual. Since conjugacy classes in
are well known, the question is which pairs are fused by conjugacy in G. Let ξ be a primitive element of
and let ζ be a primitive
root of unity in an extension
of F of degree two. Further, let B1 and
be the elements of
obtained from the identity matrix by replacing the top right 0 with 1 and with some
respectively. Our calculations will, in spirit, be similar to those in [Citation16, Lemma 4.7].
A nonidentity element is conjugate in H to either
or to one of u = B1 and
or to
for some θ that is a power of ξ or ζ; observe that one always has
even though
In the first three cases the elements w, u and
generate a single conjugacy class each. Since the element
for i a primitive 4th root of unity in F conjugates
to
and obviously
the elements
generate
distinct conjugacy classes for
and
distinct conjugacy classes for
The corresponding centralizers in G of the above five types of elements have orders
q2, q2,
and
respectively. (These facts are also included in of Section 3 displaying the irreducible characters of H.)
Table 3. Table of irreducible characters of
With the help of this we determine conjugacy classes of untwisted elements in the overgroup G of H of index two. Conjugacy in G fuses the classes generated by the untwisted elements and
into a single class (e.g. by conjugating by the twisted element
), still with centralizer of order q2, while the class generated by
remains the same under conjugacy in G, with centralizer of twice the original order, that is,
To sort out untwisted elements
observe first that the twisted element
conjugates
to
To proceed, we will use the observation from [Citation3] that two untwisted elements
and
of H are conjugate by a twisted element if and only if the traces of A and
(defined up to multiplication by −1) satisfy
This implies that two conjugacy classes in H of untwisted elements and
are fused by conjugation in G unless
itself is contained in the H-conjugacy class of
But the latter means that
and
are conjugate elements of
which is if and only if the traces of the two elements are the same up to a sign, that is,
Since taking σ-images means raising in the power of q, this equation reduces to
which is equivalent to
This happens if and only if θ is one of the
or
root of 1 or −1 in
in each case there are, respectively, q − 1 and q + 1 such roots in F. These are the situations when the G-conjugacy class of
coincides with that of
in all the remaining cases, i.e., when
the two classes are fused by conjugacy in G.
Taking this further, if a representative of a non-fused conjugacy class has been chosen for a particular θ we may automatically disregard the values
and
Note that if
then the matrices
and w are conjugate in
and if
then
represents the identity element of
Excluding the values of θ for which
it follows that in the case when
mod 4 this leaves, respectively, only
and
values of θ such that
and
with the property that the conjugacy classes of elements
considered above are mutually distinct. Similarly, if
mod 4, there are only
and
values of θ with
and
respectively, with distinct conjugacy classes of elements
as above. In both cases, one has a total of
such distinct values of θ giving distinct conjugacy classes; recall that the four 4th roots of 1 do not contribute to the classes considered here.
The set is therefore of size
observe that this number is a multiple of 8 as
mod 4. The important property of the set S is that it admits a partition into subsets of size 8 of the form
the fact that all these 8 elements are distinct follows from the way S has been defined. Let
be an arbitrary but fixed set of distinct representatives of the partition just described, so that
and for a fixed primitive element
let
Similarly, consider the set
with
which is again a multiple of 8 (exclusion of those θ for which
now reduces to leaving out ±1 only); all
are powers of the primitive
root of unity ζ introduced earlier. The set T also admits a partition into 8-element subset of exactly the same shape as before, and we let
be any fixed set of distinct representatives of this partition, with
Finally, let
Using the above facts and working out the values of θ in the case as powers of ξ we arrive at for the
conjugacy classes of untwisted elements of G.
Table 1. Table of conjugacy classes of untwisted elements of
We proceed by determining conjugacy classes of twisted elements in By [Citation3, Propositions 5 and 6], every twisted element
is conjugate in G to an element of the form
such that
or
for some
Conjugacy of twisted elements of G in the overgroup
was further investigated in detail in Propositions 7 and 8 and Theorem 1 of [Citation3]. An inspection of the proofs of the three results with emphasis on comparison of conjugacy in
and G leads to the following.
Proposition 1.
Let ξ be a primitive element of F and let be a twisted element of G. Then, exactly one of the following two cases occur:
There exists exactly one odd
such that
is conjugate in G to
with
, of order
. The stabilizer of
in G is isomorphic to the cyclic group
generated by (conjugation by) a twisted element
with
for
There exists exactly one odd
such that
is conjugate in G to
with
, of order
. The stabilizer of
in G is isomorphic to the cyclic group
generated by (conjugation by) a twisted element
with
for
Proof.
As indicated, a proof can be obtained in an almost verbatim way from the statements and the proofs of Propositions 7 and 8, and Theorem 1, of [Citation3]. For readers interested in checking the details we mark here the differences in these proofs that are significant for distinction between conjugacy in G and
Let ξ be a primitive element of F. In the cases 1 and 2 of Proposition 7, twisted elements and
with
and
for
are conjugate in
but not in G if, and only if, the ratio
in the case 1, and the product
in the case 2, have the form
for t odd. Similarly, in cases 1 and 2 of Proposition 8, elements
and
for
and
are conjugate in
but not in G if, and only if,
in the case 1, and
in the case 2, have the form
for t odd (i.e., they are non-squares in the unique subfield of F of order q). All the remaining facts in the proofs of Propositions 7 and 8 apply to conjugacy in G.
Oddness of the values of t above has the following two consequences in the proof of Theorem 1 of [Citation3] for restriction to conjugacy in G. First, the congruences appearing in parts 1 and 2 of the proof have to be taken mod and
and not just mod
and
respectively, leading to upper bounds for j in parts 1 and 2 of our proposition that are two times larger than the bounds in [Citation3, Theorem 1]. Second, parts 3 and 4 of the statement of Theorem 1 in [Citation3] refer to special cases arising due to the presence of exceptional conjugating elements in the corresponding parts of the proof. These need not be considered in our proposition, because the exceptional conjugating elements turn out to lie outside G. Again, the remaining arguments apply to conjugacy in G. □
By Proposition 1, there are a total of q conjugacy classes of twisted elements in G, consisting of classes of “diagonal type” described in part 1, and
classes of “off-diagonal type” from part 2, with centralizers of order
and
respectively. The explicit form of the twisted representatives from Proposition 1 then immediately gives our displaying conjugacy classes of twisted elements in G.
Table 2. Table of conjugacy classes of twisted elements of
3. Preliminary results on characters of ![](//:0)
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Summing up the results of and , the group splits into a total of
conjugacy classes. Note that in G the number of conjugacy classes of twisted elements, q, is by an order of magnitude smaller than the number of conjugacy classes of untwisted elements,
so that one may attempt to determine the character table of G by inducing characters from the index-two subgroup
of G. The character table of
is known and we will reproduce here a modification of the one from [Citation9, pp. 147–148], see .
The symbols B1, w,
and
used in for representatives of conjugacy classes are the same as explained at the beginning of Section 2, and
is the order of the centralizer of the corresponding element in H. The irreducible characters are ι (the trivial one), St (the Steinberg permutation character), ρ,
for
and πm for
As before, ξ and ζ are a primitive element of F and a primitive
root of 1 in
and the powers j and k in ξj and ζk are bounded by
and
Finally, in a somewhat nonstandard notation but with i denoting the complex imaginary unit as usual,
and
are complex primitive roots of unity of order
and
respectively.
Observe that has a total of
conjugacy classes. This is roughly twice the number of conjugacy classes of
for large q. An explanation offered by the previous two sections is that G-conjugation fuses “most” pairs of H-conjugacy classes of untwisted elements in H but there are only q conjugacy classes of twisted elements of G.
The degrees of irreducible characters of follow from , and those of
have been determined in [Citation16]. For convenience we display both in tabular form, with the proviso that degree 20 cannot occur in the exceptional case when q = 3.
For reference to standard concepts and results in the theory of group characters we will use the monograph [Citation7]. We will focus on results on characters of a group G with a normal subgroup H of index 2; up to the end of this section the pair G, H may be arbitrary but later we will return to our situation of G and H standing for and
The restriction χH of a character χ of G to H is the character of H defined by if
and
for
For the reverse direction of inducing characters of G from those of H, let
and
denote the centralizers of an element
in H and G. If
is a character of an index-two subgroup H of G, the corresponding induced character
of G is given as follows (for some fixed
):
We note that if
then conjugacy in G fuses a pair of distinct H-conjugacy classes
and
in H to a single G-conjugacy class (still in H); each such unordered pair
will be called a fusion pair.
For brevity we will refer to the value of the standard inner product for a character χ of G as the norm of χ, denoted by
A similar notation will be used for the norm of characters of H, and we will drop the subscript if the group is clear from the context. It is well known that χ is irreducible if and only if it has norm 1.
The following is a short summary of some results and observations of [Citation7, Chapters 5 and 6] we will need later. As usual, we let λ denote the alternating character of G, with values 1 on elements of H and −1 on elements in
Lemma 1.
Let H be a normal subgroup of G of index 2 and let be an irreducible character of H. Then,
, with
if and only if
is an irreducible character of G, and
if and only if for every fusion pair
of H-conjugacy classes in H the values of
on
are constant (i.e., if
is G-invariant); in such a case
for some irreducible character χ of G. □
Under the hypotheses of Lemma 1 this means that determination of the character table of G out of that of H reduces to understanding decomposition of characters induced by the G-invariant irreducible characters of H. This is what will be done in the next sections for and
4. Inducing characters from PSL![](//:0)
to ![](//:0)
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We begin this section by identifying the irreducible characters of G of degree 1 and q2, the unique corresponding characters of H being the trivial character ι and the Steinberg character Recall that the (irreducible) Steinberg character of a finite 2-transitive permutation group is evaluated at any permutation in the group by subtracting 1 from the number of fixed points of the permutation. The standard actions of the groups
and
on the
projective points are both 2-transitive, and even sharply 3-transitive in the case of G. It is easy to verify that the values of the Steinberg character of G, which we will denote Σ, are the same as those of
on conjugacy classes of untwisted elements, and + 1 and −1 on conjugacy classes of twisted diagonal and twisted off-diagonal elements, respectively.
Lemma 2.
Let be an irreducible character of H. If
, then
, and ι with λ are the only irreducible characters of G of degree 1. If
, then one has
Σ + Σλ, where Σ is the Steinberg permutation character, and Σ with Σλ are the only irreducible characters of G of degree q2.
Proof.
By in both cases we have for irreducible characters χ and
of G, so that Lemma 1 applies and
The rest is a consequence of uniqueness of irreducible characters of H of degree 1 and q2. □
Table 4. Degrees of irreducible characters of and
With the help of Lemma 2 and known facts from character theory we are in position to determine the number of irreducible characters of G of degrees appearing .
Lemma 3.
The number of irreducible characters of of a given degree are as follows:
Table
Proof.
By Lemma 2 we know that the number of irreducible characters of G of degree 1 and q2 is 2 in both cases. Further, by there are only three remaining degrees of irreducible characters of G, namely, and
let
respectively, be the numbers of such characters. Now,
is the number of conjugacy classes in G, and as the sum of squares of all character degrees is equal to
we also have
which can be shown to be equivalent to
(1)
(1)
Since
and
are relatively prime and the second one is odd, by the factorization appearing in (Equation1
(1)
(1) ) the number
must divide the (odd) number
and so
is a divisor of
This, however, is possible only if the two numbers are equal, that is,
which is if and only if
Having determined the value of c1 we are left with a system of two equations in two unknowns and it is easy to check that its unique solution if
and
□
We are in position to give substantial information about inducing irreducible characters from H to G. Its statement includes remarks on integrality of values of the characters, and refers to the notation used in the table of conjugacy classes of untwisted elements of (, in particular, the sets
and
) and in of characters of
the first two assertions from Lemma 2 are included for completeness.
Proposition 2.
Let be an irreducible character of
with the induced character
of
. Then, exactly one of the following cases occurs:
and
, with ι and λ being the only irreducible characters of G of degree 1; they have integral values and are non-zero on
;
and
Σ + Σλ, where Σ is the Steinberg permutation character, and Σ with Σλ are the only irreducible characters of G of degree q2; they are again integral and have non-zero values on
;
and
is one of the
irreducible characters of G of degree
such that
for
; these characters
are all real and with all-zero values on
;
and
is a unique irreducible character of G of degree
with
; it is integral and identically zero on
;
and
is one of the
irreducible characters of G of degree
such that
for
; these
are all real and all-zero on
;
and
for q – 2 pairs
of irreducible characters of G of degree
, with
for the
values
such that
, and the
values
; there is a total of
such distinct irreducible characters of G and each of these have at least one non-zero value on
Proof.
As noted, we may skip the first two items, and among the remaining ones we begin with (Equation4(4)
(4) ). If
then
is necessarily an irreducible character of G such that
is the sum of two distinct irreducible characters of H of degree
But by there are only two such characters of H, namely, ρ and
and one may check that they both induce the same character of G, giving the conclusion of (Equation4
(4)
(4) ).
Next, we turn our attention to (Equation6(6)
(6) ), where
is assumed to be reducible and of degree
By Lemma 1, we have
for some irreducible character χ of G of degree
with χH also irreducible. This means that χH must be one of the characters
for a suitable
and the same applies to
of course. But
so that by Lemma 1 the character
must be constant on any fusion pair of H-conjugacy classes. By the findings of Section 2 applied to this case, the H-conjugacy classes generated by the elements
and
for
form a fusion pair if they are distinct. The previous condition therefore means that the values of
and
on every such pair of classes must be the same (so that we may ignore distinctness here). By this translates, for a fixed
to the equality of the complex numbers
and
for all j as above.
A simple calculation (as in Section 2) reveals that the two complex numbers coincide if and only if (or, equivalently,
). To fulfill the condition on the constant value on fusion classes we are looking for the values of
for which the last equation holds for every
which happens if and only if
is a multiple of
or
In our range
given by this yields the q − 2 values of
appearing in the statement of (Equation6
(6)
(6) ), and hence also the q − 2 possibilities for
The latter give
irreducible characters χ and
of degree
all with some non-zero value on
and such that
Since the total number of irreducible characters of G of degree
is
by Lemma 3 and one of such characters was identified in part (Equation4
(4)
(4) ), the total number of irreducible characters of G referred to in (Equation6
(6)
(6) ) is
as claimed.
We continue with (Equation5(5)
(5) ), assuming that
is irreducible and has degree
The restriction of
to H cannot be irreducible, so that
is a sum of two distinct irreducible characters of H of degree
Such characters are all of the form
for suitable values of
and
in this case must also be equal to one of these. Further, our assumption on the degree of
together with Lemma 1 imply that there must be at least one fusion pair of H-conjugacy classes such that
is not constant on the pair. The fusion pairs here are generated by the same classes as in (Equation6
(6)
(6) ), and our calculations in the proof of (Equation6
(6)
(6) ) imply that a fusion pair on which
is not constant exists if and only if
is not one of the values listed in the conclusion of (Equation6
(6)
(6) ). One may check that this reduces to the condition
for the set
introduced in Section 2, with
for
(We note again that all the accompanying calculations are analogous to those in Section 2 where the condition
was considered. Here the condition on α translates to
and leads to the same conclusion that
because the expression
is invariant under the substitutions
and
the ±1 term is absorbed by the factor 4 in
)
By it may be verified that if for
then
so that
and
induce the same character of G. Referring again to and one obtains this way a total of
irreducible characters
of G whose restriction to H has the form
for
All such induced characters are distinct, and by Lemma 3 there cannot be any other irreducible character of G of degree
Finally, let us consider (Equation3(3)
(3) ), assuming that
By ,
must be an irreducible character of G, and as its restriction to H cannot be irreducible,
is a sum of two distinct irreducible characters of H of degree
The latter are all of the form πm for suitable values of m, and
must also be of this form. In Section 2, we showed that the H-conjugacy classes of the elements
and
form a fusion pair for every
One may check that the values of any given πm are non-constant on at least one of these fusion pairs, which conforms to the last part of Lemma 1. Further, a direct verification against shows that if
then
Thus, both πm and πmq determine the same induced character of G, and by and they give rise to
irreducible characters of G of the form
corresponding to the values
It may be verified that these induced characters are distinct, and by Lemma 3 there is no other irreducible character of G of degree
As regards remarks on integral and real values, most of them are obvious, and note that is always identically zero on
This completes the proof. □
An inspection of the numbers in Proposition 2 reveals that in the cases (Equation1(1)
(1) )–(Equation5
(5)
(5) ) there are, respectively,
real characters, and all of them have been completely determined. This makes a total of
irreducible characters of G that are all real. Subtracting this from the number of all irreducible characters of G leaves us only with the
ones described in part (Equation6
(6)
(6) ) of Proposition 2, that may assume non-real values, but only on
as all the characters of H are real.
We now address the question of possible character values that are not real. A well-known result in the theory of group characters is that, for an element g of a group, the values of all irreducible characters of the group evaluated at g are real if and only if g is conjugate to in the group, or, equivalently, the conjugacy class of g is closed under inversion in the group; such conjugacy classes are called real. By every conjugacy class of untwisted elements in
is real. For conjugacy of twisted elements we have, in the notation of Proposition 1:
Lemma 4.
The conjugacy class of the twisted element in
for odd j,
, is real if and only if
mod 4 and
. The conjugacy class of
for odd j,
, is real if and only if
mod 4 and
Proof.
Let and
Observe first that
and
By Proposition 1 the elements d(j) and
are in the same conjugacy class in G for odd j,
if and only if
mod
This can only happen if
mod 4, and then the congruence is equivalent to
mod
As the modulus of the last congruence is relatively prime to
it follows that this only leaves us with the value
in our range for j. The analysis for the elements
and
is analogous. □
This gives, in both cases mod 4, exactly q − 1 conjugacy classes of twisted elements in G that are not real. By the earlier remarks we also know that there are at most irreducible characters of G (those from part (Equation6
(6)
(6) ) of Proposition 2) that are not real.
5. Representations for the remaining characters
In the previous section, we have almost completely determined the character table of and we have been left with
“missing” irreducible characters, which are the only ones that may assume non-real values on q − 1 conjugacy classes of G formed by elements of
These characters have been referred to in part (Equation6
(6)
(6) ) of Proposition 2 and from now on we will denote them by
and
with
for the total of q − 2 values of
in the set
, where
and
We will determine these characters in the next section; here we first derive some related representations by extending one-dimensional representations of a suitable subgroup of H. The method is an adaptation of derivation of principal series representations for two-dimensional special linear groups (cf. [Citation2, p. 232]).
Let be the subgroup of H < G,
induced by upper-triangular matrices with determinant 1. Explicitly, if
is a fixed primitive element as before and if h(u, d) is the 2 × 2 matrix with first and second row of the form
and
for an arbitrary non-negative integer
and any
then
(2)
(2)
observe that
It is well known (see e.g. [Citation2, p. 232] adapted to the projective case) that for every
(and, in fact, for every integer
but this will not be needed here) the assignment
(3)
(3)
defines a one-dimensional non-real representation of
To extend such a representation to one of the entire group
we proceed as ibid by first constructing a suitable set of coset representatives of
in G. To do this, for
we introduce 2 × 2 lower-triangular matrices m(t) and
as well as matrices
and
as follows:
(4)
(4)
Using the
matrices from (Equation4
(4)
(4) ) we introduce
elements of G by letting
note that
is the unit element of G. It may be checked that this set of
elements of G is a left transversal for the subgroup
With the help of this transversal we will now extend any one-dimensional representations of
described in (Equation3
(3)
(3) ) to an nq-dimensional representation
of G; the method originates from [Citation5]. Before doing so we will make an agreement about indexation. Let
where the entries
are considered mod
and let
To describe
matrices we will use the nq indices from the set
equipped with the linear ordering
(5)
(5)
Invoking now [Citation2, Lemma 9.1] adapted to our situation and using the introduced notation, an nq-dimensional representation is obtained by assigning, to every element
the
matrix
whose (a, b)th entry for
is determined by the following rules (using the convention that
for our indices):
By part (Equation6(6)
(6) ) of Proposition 2, each induced representation
for
is reducible and splits into two irreducible representations of dimension
each. The missing irreducible characters
and
on
for
are given by traces of these representations, or, equivalently, by traces corresponding to the two G-invariant
-dimensional subspaces of the representation
For evaluation of these traces (which we will do in the next section) we will need explicit knowledge of -images of a generating set of
and its subgroup
It is well known that
is generated by the two elements
and
of order
and 3, respectively. In accordance with , as representatives of conjugacy classes of
we will take the elements
and
for odd positive integers
and
respectively. For the
-images of these elements we obtain:
Proposition 3.
For every the values of the induced representation
at the elements g1, g2, g3, and g4 are
unitary matrices with entries as follows, where
, and indices in
distinct from
are understood mod
:
If
, then
If
, then, letting
where the function f on residue classes t mod
and
is given by
If
for an odd positive integer
, then
If
for an odd positive integer
, then
Proof.
For illustration we will only deal with the entire part (b) and the second items of both (c) and (d), as the verification in all the remaining cases is analogous; the matrices are unitary by inspection and hence so is the entire induced representation For all calculations we note that
was introduced before the statement of Proposition 3, preceded by an exposition of the associated matrices in (Equation4
(4)
(4) ) together with the elements xa and yb.
Part (b). Letting for
one has
if
otherwise
By the rules (explained in Section 1) for calculations in our group G, the inverse of
is
where n(t) is the 2 × 2 matrix with rows
and
Following these rules (and slightly abusing the notation and using
also as exponents at ξ), the product
evaluates to
The condition
is equivalent to
which, for
determines b as a function f(a) from the equation
In such a case the main diagonal of X consists of the elements
and
so that by (Equation3
(3)
(3) ) the value of
is equal to
One may check that for the pair
evaluation of the product
reduces to determining membership of
in
and gives the same condition on a, b as above, that is,
for
with the same value
of
Calculating all of
and their y-versions one obtains matrices with main diagonal 1, 1, implying the second item of (b).
Regarding the function f, mutual equivalence of the three equations and
implies that f(a) = b,
and
for
Since raising in qth power is an automorphism of
one also has
It follows that if q is not a power of 3 then f as a permutation of the undashed indices not in
has two fixed points, namely, the two primitive 3rd roots of 1 in F, and for the remaining values, f consists of 3-cycles of the form (a, b, c) where
and
if q is a power of 3 then f has 0 as its unique fixed point and the remaining orbits of f are 3-cycles as above. An analogous statement applies to dashed indices.
The second item of part (c). For finding the value of
by the calculation rules in G reduces to evaluating the product
which results in
Here one has
if and only if
mod
To evaluate
at such an X one needs to represent it in the form
which reduces to looking for an element
such that zX has a pair of mutually inverse entries ξu and
in the main diagonal. The obvious choice here is
giving by (Equation3
(3)
(3) ) the value
The second item of part (d). Here, in the product X of the three matrices is
Now, X belongs to
if and only if
mod
and to determine the corresponding value of
we again need to represent this element as
by finding some
such that the main diagonal of zX has the form
Using
and
multiplication by
produces the required diagonal entries in zX for
By (Equation3
(3)
(3) ), the value of
at
for
is then equal to
Finally, with the help of
and the substitution for u the value of
simplifies to
which is as claimed if a = t and
□
It will be of advantage to restate the above results in form of block matrices. For let
be the matrix of dimension
indexed by the ordered set
with
and
for every
mod
and with all the remaining entries equal to zero. We will leave out the subscript
if no confusion is likely. Further, let
B,
and
for odd positive j be square matrices of dimension
indexed by the ordered set Ind, and defined as follows:
(6)
(6)
where entries of B not listed are assumed to be zero; the parameter j will be omitted if no loss of clarity is likely. From now on we will also extend the symbols
and
also to 2 × 2 blocks of dimension
One may check that, in this new notation, the matrices
and
from Proposition 3 can be displayed in the form
(7)
(7)
where we assume a natural extension of indexation from Ind to Ind′ in the bottom
coordinates of the “larger” matrices of dimension
this also justifies the usage of the subscripts “upp” and “low” for the upper and lower non-zero blocks of the matrices of (Equation6
(6)
(6) ) which appear in (Equation7
(7)
(7) ).
Since the matrices and
of dimension
assigned to the generators g1 and g2 of
in the representation
are block-diagonal, it follows that all matrices of
assigned to elements of H are block-diagonal (with diagonal blocks of dimension
). Thus, the restriction
of the unitary representation
to H is a sum of two irreducible
-dimensional unitary representations, say,
and
of the group H; formally,
This is a good point to recall that our aim is to determine the characters denoted
and
at the beginning of the previous section (which are the characters of part (Equation6
(6)
(6) ) of Proposition 2) for
with
As the restrictions
and
coincide and are equal to the character
of
it follows that in the sum
the constituents
and
generated, respectively, by the pairs
and
must be equivalent unitary irreducible representations of H. This implies the existence of an “intertwining” unitary matrix
of dimension
with the property that
for every
In particular, using the common notation
for conjugation, for the constituents of
and
in (Equation7
(7)
(7) ) one has
(8)
(8)
Further, using
from (Equation6
(6)
(6) ) and realizing that
one obtains
Since from (Equation8
(8)
(8) ) we obviously have
it follows that M2 commutes with the representation
By Schur’s theorem, M2 is a constant multiple of the identity matrix. However, for the purpose of intertwining the constant multiple may be an arbitrary non-zero complex number, which we will henceforth choose to be equal to 1. Thus, without loss of generality we may assume that
which means that M is a unitary Hermitian matrix; now we also have
6. Determination of the remaining characters
We begin this section by determining all the non-trivial G-invariant (and hence -dimensional) subspaces of our
-dimensional representation
The method relies on the following result which extends [Citation15, Lemma 2.6, p. 56] and is likely to be folklore in representation theory; we therefore include only a short proof.
Lemma 5.
Let be a direct sum of a pair of equivalent non-real irreducible representations of a group K with associated disjoint vector spaces V1 and V2 and with an intertwining matrix N. If a non-trivial subspace V of
is K-invariant, then either the projection of V onto exactly one of V1, V2 is zero, or there is a non-zero
such that
Proof.
By default, all of and Vi for i = 1, 2 as well as V and N must have the same dimension. Letting πi be the projection of
onto Vi for i = 1, 2, by irreducibility the intersection
is either Vi or trivial. Leaving the possibility when (exactly) one of
is equal to Vi we are left with the case when both intersections
are trivial. But then, letting
denote the restriction of πi to V, it follows that
for i = 1, 2; in particular, the mappings
are K-invariant isomorphisms. The composition
is a K-invariant isomorphism
which, by Schur’s theorem, must be given by
for an intertwining matrix N, unique up to a non-zero multiplicative constant. Now, if
is an arbitrary vector, letting
we obtain
□
Applying Lemma 5 to our restricted representation of the subgroup H < G one sees that an invariant subspace W for
has either zero projection onto the first (undashed)
coordinates in the set
or a zero projection onto the second (dashed) coordinates in
or else has the form
where
is the intertwining involutory matrix of (Equation8
(8)
(8) ). One of these then must form an invariant subspace for the entire representation
of the group G. Realizing that the representation
is obtained from
by adjoining any of the matrices
from (Equation7
(7)
(7) ) with zero diagonal blocks, the first two possibilities are immediately ruled out, and we obtain:
Corollary 1.
Every non-trivial invariant subspace of has the form
, where M is the intertwining involutory matrix of the representations
and
of H generated by
and
, and c is a non-zero complex constant. □
By the corollary, for every vector i.e., of the form
for
the vector
must also belong to W. This means that
and as
was arbitrary and
it follows that
(9)
(9)
With the help of Corollary 1 and the knowledge of and
by (Equation6
(6)
(6) ) we now determine the possible values of the constant c. Namely, squaring the expressions for
and
one obtains
and substituting these into the square of (Equation9
(9)
(9) ) gives
The last equation reduces by (Equation8
(8)
(8) ), i.e., by
to
so that
for some
where i is the complex imaginary unit. Hence, (Equation9
(9)
(9) ) can equivalently be written in the form
(10)
(10)
Observe that if c is one of the four values determined above for which the subspace is G-invariant, then
is another such subspace and the pair
forms an orthogonal decomposition of
Indeed, letting
denote the complex conjugate transpose and using the fact that M is unitary (
) together with
evaluating the standard inner product of (complex) vectors from W and
gives
Consider such a pair of G-invariant subspaces. From the way our induced representation
was introduced before the statement of Proposition 3 and from the calculations in its proof it follows that for every
the unitary matrix
has the form
with off-diagonal (and necessarily unitary) blocks of dimension
each. Consider a complex eigenvalue x of
associated with a non-zero eigenvector
splitting uniquely as
for
and
Now,
and by
and the G-invariance of the two subspaces we obtain
and
It follows that the eigenspace
of
associated with the eigenvalue x is a direct sum
of the corresponding spaces of eigenvectors of
that belong to W and
By diagonalizability of unitary matrices this also means that the spectrum of
of size
is a concatenation of the spectra
and
of size
consisting, respectively, of the eigenvalues corresponding to eigenspaces contained in W and
Let now be an eigenvector of
for an eigenvalue x. Since W is G-invariant and hence preserved by
one has
Membership of the last vector in W means that
and since this holds for every eigenvector
corresponding to any eigenvalue x, we obtain
or, equivalently,
The chain of displayed equations further gives
and
which means that
and
are eigenvectors of the matrices
and
respectively, for the same eigenvalue x. Conversely, if
is an eigenvector of
for an eigenvalue x, then right multiplication by cM shows that
is an eigenvector of
for x and hence
that is,
is an eigenvector of
for the same eigenvalue x.
This correspondence between the eigenspaces of that are subspaces of W and eigenspaces of the (unitary and hence diagonalizable) matrix
together with the earlier established facts about spectra lead to the conclusion that the multi-set
is equal to the spectrum of
This way we have arrived at the following conclusion for our missing character
for
which we state as a summary of the above considerations together with (Equation8
(8)
(8) ) and (Equation10
(10)
(10) ).
Proposition 4.
Let M be a unitary Hermitian matrix of dimension with determinant ±1 such that
(11)
(11)
for some
. For an arbitrary
let
. Then, the pair of characters
and
for each
is determined by letting
for every
□
Proposition 4 allows for a quick determination of the missing pair of characters and
for
The key observation now is that for
(
) one has
by Proposition 3. Let Q be the permutation matrix corresponding to the permutation of the set Ind fixing * and
and sending
for every
mod
Observe that, due to
for the matrix P introduced immediately after the end of the proof of Proposition 3 one has
With this in hand and using oddness of j, an inspection of (Equation6
(6)
(6) ) shows that
(12)
(12)
Comparing (Equation12
(12)
(12) ) with (Equation11
(11)
(11) ) implies that for
one can simply take M = Q for the intertwining matrix, with
and
Applying Proposition 4 further, for the value of the “missing” character
at the element
of part (c) of Proposition 3 may be taken to be the trace of the product
By (Equation6
(6)
(6) ) one has
and one may check that the only non-zero diagonal elements of the product
are those in positions * and
which are
and
Therefore
for
By the same token, letting
for the element
of part (d) of Proposition 3, for
one has
and as this matrix has a zero diagonal by inspection it follows that
Using the notation for
with
the missing pair of characters
may be given as follows.
Corollary 2.
For , the pair of characters
and
are determined by
for odd positive
, and
for odd positive
□
We continue by calculating the values of for
on conjugacy classes of twisted elements of G. Out of the previous results it is easy to extract, for each of the
values of
the
-dimensional vector of values of
Observe that, for any fixed
the restriction
is orthogonal to each of the
restrictions
for
and to the trivial character ι. It follows that, for our fixed
the
-dimensional vector
is orthogonal to the system of
mutually orthogonal vectors consisting of the all-one vector and the
vectors
for the
values of
all of dimension
But such a
must then be the zero vector, which implies that
is zero at all elements
for every
It remains to determine the values of for
at the elements
for odd positive
which will take considerably more space. To this end it is sufficient to determine the matrix M for this situation, and although general methods for calculating intertwining matrices are available (see e.g. [Citation11]) we adopt here a more direct approach. Summing up some of the facts established so far, the unitary Hermitian matrix M with
is, up to the sign, determined by the equations from Proposition 4, the second and third of which are equivalent to BM = MB and
Written in terms of coordinates
for
the first equation gives
(13)
(13)
To deal with the second equation we will assume that
and we will choose
such that
as we shall see, this choice will be consistent with the forthcoming calculations. Taking this into the account and realizing that now
it can be checked that the equation
taken first for j = 1 and then extended by induction for every t mod
translates into the following linear system, where
(14)
(14)
It can also be verified that the system (Equation14
(14)
(14) ) for t = 2 implies the first equation
from Proposition 4.
In what follows we will use the fact that is the only non-trivial involution mod
with
mod
we will also frequently use the cycle
of the permutation f. Let
The leftmost equation in the last row of (Equation14
(14)
(14) ) shows that
for every
Applying first (Equation13
(13)
(13) ) with
followed by using the middle equation of (Equation14
(14)
(14) ) for t = q and finally by (Equation13
(13)
(13) ) again one obtains
But from
it also follows that
which in combination with the above implies
that is, y is a real number. Using (Equation13
(13)
(13) ) it also follows that
so that the entries of the column of M marked
are constantly equal to y except for
Due to
the same conclusion is valid for the row of M marked
By
the dot product of the row and the column of M marked
must be equal to 1, that is,
from which
The entries of M in the row and column marked * are determined by the equations of (Equation14(14)
(14) ) containing asterisks. To determine the remaining entries, observe first that, by (Equation14
(14)
(14) ) and (Equation13
(13)
(13) ), one has
and hence by the first equation of (Equation14
(14)
(14) ) one also has
for every t mod
The remaining entries of M turn out to be non-zero and can be determined as follows. For every
by (Equation13
(13)
(13) ) one obtains
Combining this with application of the first equation of (Equation14
(14)
(14) ) yields
(15)
(15)
Letting now
and
and evaluating t and
in terms of a and b gives
and
and hence (Equation15
(15)
(15) ) with
reduces to
(16)
(16)
the condition
is a consequence of
Now, (Equation16
(16)
(16) ) together with the information about zero entries of M and about rows and columns marked * and
determine the matrix
completely. Conversely, one may check that the entries of
that have just been determined satisfy EquationEquations (Equation13
(13)
(13) )
(13)
(13) and Equation(Equation14
(14)
(14) )
(14)
(14) with
With this in hand we are in position to calculate the remaining values of for
at the elements
represented by the matrix
given in part (d) of Proposition 3; in particular, entries of
are zero except for those indexed
and
for
equal, respectively, to
and
By Proposition 4 we have
and a straightforward evaluation of the trace gives
With the help of (Equation16
(16)
(16) ) and
and also by changing the summation variable from a to
the trace equation transforms to
(17)
(17)
Again the key is to use the fact that now
so that to understand the sum in (Equation17
(17)
(17) ) one just needs to study the residues of
mod
and it is sufficient to do this only for
as the values of
depend only on the residue of t but this time mod
The following auxiliary result will help sort the situation.
Lemma 6.
For the values of
are in distinct congruence classes mod
; moreover,
and
mod
Proof.
Let Suppose that for
one has
and
for the same residue v mod
Observe that the power
of the primitive element
that has been used throughout is an element of the subfield
of F. By the defining equation for f in part (b) of Proposition 3 one then has
and, similarly,
Eliminating ξv from the two equations gives
But as
for
while all the powers of z are in F0, it follows that u1 = u2 and hence also t1 = t2, proving the first part of the result. The second part is proved similarly, letting e(t) = j and then e(t) = 0 in the first part of the calculation above. □
The set contains, for each
exactly q + 1 distinct elements with residue mod
equal to u, and each of these q + 1 elements of the form
for
determine the same value of the argument of
in (Equation17
(17)
(17) ). Therefore the sum in (Equation17
(17)
(17) ) evaluates to
(18)
(18)
By Lemma 6, the values of
appearing in (Equation18
(18)
(18) ) cover exactly q − 1 out of the possible q + 1 residue classes mod
except for the classes 0 and j. But the sum of powers of
with exponents ranging over all the q + 1 distinct residue classes mod
is equal to zero, because it is precisely the sum of all the q + 1 distinct complex
roots of unity. It follows that the sum of (Equation18
(18)
(18) ) is equal to the negative of the “missing” two terms corresponding to residue classes j and 0, that is,
(19)
(19)
Substituting now (Equation19
(19)
(19) ) into (Equation18
(18)
(18) ) and then the resulting sum into (Equation17
(17)
(17) ) finally gives the values of
equal to
We state the corresponding result as a corollary, using the notation
for
where
we will also take into account the earlier observation about zero values of
for
Corollary 3.
For , the pair of characters
and
are determined by
for odd positive
, and
for odd positive
□
7. The character table of ![](//:0)
![](//:0)
It remains to translate the facts derived in Propositions 2 and Corollaries 2 and 3 into a tabular form. In the interest of saving space and displaying all the irreducible characters of in form of a single table one needs to use a number of abbreviations. The following is a legend for reading .
Table 5. Table of irreducible characters of
Characters. We use the symbols ι, ρ, and in the first column of in their meaning as in of characters of
and Σ for the Steinberg character. In the explanations below we will also use the characters
and πm of
Continuing in the description of character designations in the first column of , we let
for
and
for
the sets
with
and
with
have been introduced in Section 2 before . The last two entries in the first column of are based on the characters
from Corollaries Citation2 and Citation3, and we let
for
and
for
In rows marked
for
Σ,
where double signs appear on some values, those with the bottom signs are assumed to be values of the product of ψ with the alternating character λ. The number of characters totals to
which is the number of conjugacy classes as stated at the beginning of Section 3.
Conjugacy classes. Again, in the interest of saving space, in displaying representatives of conjugacy classes of in the first row of we will omit square brackets, and the second coordinate, 0 or 1, will appear as a subscript; thus, for example, instead of
and
we simply write
and
The second and the third row of display the number of conjugacy classes (
cl) and orders of the corresponding centralizers (
); the size of a conjugacy class can be obtained by dividing the order
of
by the order of the centralizer. The column
corresponds to those representatives
for which
The column headed by
actually comprises four columns, namely, those corresponding to the total of q − 2 values of
and
from , including the bounds and restrictions on j as given in this table. (Observe that although entries in these columns have the same shape, they take different values of j and
as input.) The column headed
applies for
and the in the last two columns corresponding to
and
the variable j is odd and bounded by
and
respectively.
Entries. We have used the following in : where
where
for
where
and finally
for
where
(Note that the complex numbers α and β have also been used in , the character table of
)
In the terms introduced above we are finally in position to present the character table of the twisted linear group
Theorem 1.
The character table of the group is given by .
In a forthcoming paper [Citation10], we apply the character table from Theorem 1 to obtain an enumeration of regular maps of any given type (i.e., of given vertex valency and face length) supported by the groups This refines the earlier enumeration of [Citation3] where types were not considered, and extends the available collection of classes of regular maps of given type [Citation8] enumerated with the help of Frobenius’ character formula [Citation4].
Acknowledgments
The authors thank the anonymous referee for pointing out an inaccuracy in the original handling of conjugacy classes and for very helpful and illuminating comments and suggestions that led to improvements in the presentation of our results.
We also thank G. A. Jones for helpful discussions and G. Erskine for computational results that enabled to verify validity of the character table of for
Additional information
Funding
References
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