Orthogonal determinants of SL3(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{SL}\,}_3(q)$$\end{document} and SU3(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{SU}\,}_3(q)$$\end{document}

We give a full list of the orthogonal determinants of the even degree indicator ‘+’ ordinary irreducible characters of SL3(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SL}_3(q)$$\end{document} and SU3(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SU}_3(q)$$\end{document}.


Introduction
Let G be a finite group and ρ : G → GL n (C) be a representation.We call ρ an orthogonal representation, if there is a symmetric, non-degenerate, ρ(G)-invariant bilinear form β on C n .It is well-known that being an irreducible orthogonal representation is equivalent to the associated character having Frobenius-Schur indicator '+', i.e. ρ being equivalent to a real representation.A character is called orthogonal if it is the character afforded by an orthogonal representation.The character χ is orthogonal, if and only if it is of the form where k ) are irreducible characters of G with Frobenius-Schur indicator '+' (resp.'-', resp.'0'), and a i , b j , c k are non-negative integers.
Let χ be an orthogonal character as in equation (1) and let K = Q(χ) be the character field of χ.The main result of [8] shows that if the degree of all χ + i is even then there is a unique element det(χ) := d ∈ K × /(K × ) 2 , called the orthogonal determinant of χ, such that for all representations ρ : G → GL n (L) over a field extension L ⊇ K affording χ all non-degenerate, ρ(G)-invariant, symmetric bilinear forms β on L n have the same determinant We call such a character orthogonally stable.
The orthogonal determinant of 2χ (−) j is always 1 (see [13]) and for characters of the form χ 0) the orthogonal determinant can be obtained from the character values as given in Lemma 2.4.So it remains to deal with the indicator '+' part.Put In a long term project theoretical and computational methods are used to calculate the orthogonal determinants of the small finite simple groups (see [2] for a survey).The goal of this paper is to determine the orthogonal determinants of the characters in Irr + (G) for the two infinite series of finite groups of Lie type, G = SL 3 (q) and G = SU 3 (q), for all prime powers q.For SL 2 (q) the orthogonal character table is already computed in [1].An important subgroup to analyse the structure and the representations of a finite group G of Lie type is its standard Borel subgroup B. A character χ ∈ Irr + (G) is called Borel stable, if the restriction of χ to B is orthogonally stable.The structure of B as a semidirect product of a p-group and a split torus T allows us to determine the orthogonal determinants of all Borel stable characters of G (see Remark 4.2).For the groups G = SL 3 (q) and G = SU 3 (q) it turns out that those χ ∈ Irr + (G) that are not Borel stable occur nicely in a permutation character.For such characters Lemma 2.5 can be used to determine their orthogonal determinants.
This paper is a contribution to Project-ID 286237555 -TRR 195 -by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

Methods
This section collects some basic results about orthogonal determinants.
The paper [7] gives an easy formula for the orthogonal determinant of orthogonally stable characters of p-groups.We only need the following special case: [7,Corollary 4.4]) Let p be an odd prime and let χ be an orthogonally stable rational character of a finite p-group.Then p − 1 divides χ(1) and For the cyclotomic fields Lemma 2.5.Let G be a finite group acting on a finite set M. Let V be the permutation representation over Q. Define the G-invariant bilinear form β : from which the result follows.
3 The Orthogonal Characters of SL 3 (q) and SU 3 (q) Let p be a prime and let q be a power of p.The group SL 3 (q) is Let F q 2 ⊇ F q be the field extension of degree 2 and let F : F q 2 → F q 2 , x → x q be the Frobenius automorphism.For a matrix A ∈ F n×m q 2 we define F (A) to be the matrix where we apply F component-wise.Let The letter G will always denote one of SL 3 (q) or SU 3 (q).The full character table of G was first calculated in [11] in 1973 by Simpson and Frame.By 'Ennola duality' (see [4] for the statement and [6] for a proof), "SU 3 (q) = SL 3 (−q)", the irreducible characters of SU 3 (q) can be obtained from the ones of SL 3 (q) by formally replacing every instance of q by −q, so that there is a single generic character table giving the character table for both groups introducing an additional parameter ε = +1 for G = SL 3 (q) and ε = −1 for G = SU 3 (q).
In this notation the center of G is the group of scalar matrices in G and hence of order gcd(q − ε, 3).In particular the set Irr + (G) is the set of irreducible orthogonal characters of even degree of the group PSL 3 (q) = SL 3 (q)/Z(SL 3 (q)) and PSU 3 (q) = SU 3 (q)/Z(SU 3 (q)).
It is well known that, with the exception of PSU 3 (2), the groups PSL 3 (q) and PSU 3 (q) are simple groups for all q.The irreducible characters of PSU 3 (q) and PSL 3 (q) are the irreducible characters of G that are constant on the center.
Gow [5] showed that all the characters of PSL 3 (q) and PSU 3 (q) have Schur index 1, except the unique character of degree q 2 − q of PSU 3 (q), which has Schur index 2 and Frobenius-Schur indicator '-'.Additionally, the results in [10] allow us to obtain the character fields from some combinatorial description.For cyclotomic numbers we use the notation from Lemma 2.4 and for the naming convention of the irreducible characters we follow [11,Table 2].Then the set Irr + (G) is given as follows: Theorem 3.1.The following table includes all χ ∈ Irr + (G), their degrees χ(1) and character fields Q(χ): • For q odd and G = SL 3 (q), Irr + (G) = {χ qs , χ • For q odd and G = SU 3 (q), Irr • For q even and G = SL 3 (q), Irr + (G) = {χ qs , χ q 3 }.

Note that the characters χ (u)
st ′ only exist for 3|q − ε.

Results
Let G = SL 3 (q) or G = SU 3 (q).Let Then U is the unipotent radical of B and a Sylow p-subgroup of G, and B = N G (U) = U ⋊ T is a (standard) Borel subgroup, where T := {diag(d, e, f ) ∈ G} is a maximal torus.Denote by W = N G (T )/T the Weyl group of G.
We need an explicit notation for Irr(T ): For G = SL 3 (q) we fix a generator t of F × q .Then the torus For G = SU 3 (q) the torus is isomorphic to F × q 2 =: τ .So T = {τ a := diag(τ a , τ (q−1)a , τ −qa ) | a ∈ {0, . . ., q 2 − 2}} and To unify notation we put where χ T is the character of T on the U-fixed space, also known as the Harish-Chandra restriction of χ.In particular its degree is χ Note that T is abelian and so χ T is a sum of linear characters.If these characters are complex (i.e. of indicator '0') then χ T is orthogonally stable and its determinant can be computed from Lemma 2.4.In fact the irreducible constituents of χ T can be obtained from the action of the Weyl group W on Irr(T ).It is well-known that Let θ ∈ Irr(T ).Then θ can also be considered as a character of B. A character χ ∈ Irr(G) is said to be in the principal series if χ appears in Ind G B (θ) for some θ ∈ Irr(T ).
We are now ready to give the main result.

Proposition 4 . 6 .
Let G = SU 3 (q).The only characters in Irr + (G) which are not in the principal series are χ (u,−u,0) st and χ (u) st ′ for q odd.
−u,0) st ) = det((χ (u,−u,0) st ) T ) det((χ (u,−u,0) st Definition 2.1.Let G be a finite group, H ⊆ G a subgroup, and let χ be an orthogonal character of G. Then χ is called H-stable if the restriction Res G H (χ) of χ to H is an orthogonally stable character of H.

)
Remark 4.1.By Lemma 2.2 (i) the orthogonal determinant of a B-stable character χ of G is determined by the restriction of χ to B. Decompose this restriction as