ON THE EXISTENCE OF A GROUP ORTHONORMAL BASIS. Peter Zizler (Received 20 June, 2014)

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 45 (2015, ON THE EXISTENCE OF A GROUP ORTHONORMAL BASIS Peter Zizler (Received 20 Jue, 2014 Abstract. Let G be a fiite group ad let l 2 (G be a fiite dimesioal Hilbert space of all complex valued fuctios for which the elemets of G form the (stadard orthoormal basis. We say a set of fuctios {α i } i l 2 (G is G- orthoormal if α i (tα j (tτ = α i (τδ i,j for all τ G. I our paper we prove that l 2 (G admits a G-orthoormal basis if ad oly if G is a abelia group. Moreover, if G is o-abelia tha the size of the largest G-orthoormal set i l 2 (G is the sum of the degrees of the irreducible represetatios of G. 1. Itroductio Cosider a fiite group G ad let l 2 (G deote the fiite dimesioal Hilbert space of all complex valued fuctios, with the usual ier product <. >, for which the elemets of G form the (stadard basis. We assume that this basis (G is ordered ad make the atural idetificatio, as vector spaces, with C, where G =. We say that a set {α i } i l 2 (G is G-orthoormal if α i (tα j (tτ = α i (τδ i,j for all τ G. I the case of a cyclic group G, G =, the complex expoetials {α j }, defied by α j (k = e 2πijk, form a G-orthoormal basis for l 2 (G. I fact, up to a multiplicatio by a complex umber of modulus oe, this is a uique G-orthoormal basis for l 2 (G. These complex expoetials yield a frequecy decompositio of l 2 (G with the value j referred to as the associated frequecy. I the group theory laguage, these complex expoetials are the multiplicative characters for the cyclic group G. It is ot immediate how to geeralize the cocept of a frequecy decompositio to l 2 (G where the group G is o-abelia. The complex expoetials are the multiplicative characters for the cyclic group G ad this is the key property that makes them the cadidates for the frequecy decompositio fuctios. I the oabelia case the multiplicative characters are still good cadidates for the aalogue 2010 Mathematics Subject Classificatio Primary 43A32, Secodary 42C99. Key words ad phrases: G-orthoormal basis, o-abelia Fourier trasform, o-abelia covolutio, irreducible represetatio.

2 46 PETER ZIZLER of the frequecy decompositio fuctios, however, i the o-abelia case we do ot have eough of them. Our approach here will be to geeralize the otio of G-orthoormality from the cyclic case to the o-abelia case. After all, what makes the complex expoetials good cadidates for a frequecy decompositio is their G-shift ivariace (cyclic o the correlatio amog them. Of course, istead of cyclic shifts, we are goig to have geeral coset shifts i the o-abelia case. Now let G be a o-abelia group. Our aim ow is to costruct a G-orthoormal set i l 2 (G as large as possible, ideally a basis for l 2 (G. However, we will show that havig a full basis for l 2 (G cosistig of G-orthoormal fuctios is impossible i the o-abelia case. It turs out that the largest set of G-orthoormal fuctios i l 2 (G has m elemets, where m is the sum of all the degrees of irreducible represetatios of G. However, if we relax the coditio for G-orthoormality to just G decorrelatio, where α i (tα j (tτ = 0 for all τ G ad i j the such a (G decorrelated basis exists for l 2 (G. The key to our aalysis is to coect the cocept of G-orthoormality with the cocept of a o-abelia covolutio which i tur will allow us to employ the o-abelia Fourier trasform. A G-covolutio of ψ ad φ, where ψ, φ l 2 (G, is defied by the followig actio, σ G (ψ φ(σ = τ G ψ(στ 1 φ(τ. Let C deote the complex umbers. A fiite dimesioal represetatio of a fiite group G is a group homomorphism ρ : G GL(d j, C where GL(d j, C deotes the geeral liear group of degree d j, the set of all d j d j ivertible matrices. We refer to d j as the degree of the group represetatio. Two group represetatios ρ 1 : G GL(d j, C ad ρ 2 : G GL(d j, C are said to be equivalet if there exists a ivertible matrix T M dj d j (C such that for all g G. T ρ 1 (g T 1 = ρ 2 (g Every fiite dimesioal group represetatio is equivalet to a represetatio by uitary matrices, see [4]. For more iformatio o group represetatios see [1], [2] or [3].

3 ON THE EXISTENCE OF A GROUP ORTHONORMAL BASIS 47 Let Ĝ be the set of all (equivalece classes of irreducible represetatios of the group G. Let ρ Ĝ be of degree d j ad let φ C. The the Fourier trasform of φ at ρ is the d j d j matrix φ(ρ = s G φ(sρ(s. The Fourier iversio formula, s G, is give by φ(s = 1 G ρ j Ĝ ( d j tr ρ j (s 1 φ(ρ j. We alert the reader to a ivolutio switch s s 1 i the summad fuctios. We refer the reader to [4] for more details. Let C[G] be the algebra of complex valued fuctios o G with respect to G-covolutio. Let ψ = (c 0, c 1,..., c 1 C ad idetify the fuctio ψ with its symbol Ψ = c c 1 g 1 + c 1 g 1 C[G]. Let ψ ad φ be two elemets i C. We have a atural idetificatio ψ φ ΨΦ uderstood with respect to the iduced group algebra multiplicatio. We have a o-abelia versio of the classical z trasform. The actio of ψ o φ through G-covolutio is captured by the matrix multiplicatio by the G-circulat matrix C G (ψ, i particular ψ φ = C G (ψφ. The character of a group represetatio ρ is the complex valued fuctio defied by χ : G C χ(g = tr(ρ(g. A character is called irreducible if the uderlyig group represetatio is irreducible. We defie a ier product o the space of class fuctios, fuctios o G that are costat its cojugacy classes χ, θ = 1 G χ(gθ(g. g G Note that a character is a class fuctio. We have as may irreducible characters as there are cojugacy classes of G. If G is abelia, the we have irreducible characters. With respect to the usual ier product we have χ i, χ j = δ ij where δ i,j is the Kroecker delta. Irreducible characters form a basis for the space of class fuctios o G, see [1]. The Fourier trasform gives us a atural isomorphism

4 48 PETER ZIZLER where C[G] M(Ĝ M(Ĝ = M d 1 d 1 (C M d2 d 2 (C M dr d r (C with d d d 2 r =. A typical elemet of C is a complex valued fuctio ψ = (c 0, c 1,..., c 1 ad the typical elemet of M(Ĝ is the direct sum of Fourier trasforms φ(ρ 1 φ(ρ 2 φ(ρ r. Fourier trasform turs covolutio ito (matrix multiplicatio ψ φ = r ψ j φj = ψ φ j=1 I the abelia settig the Fourier trasform is a uitary liear trasformatio (proper scalig required. I the o-abelia settig we recapture this property if we defie the right ier product o the space M(Ĝ. Let φ C ad defie for s G φ j (s = d ( j G tr ρ j (s 1 φ(ρ j. Note φ = r j=1 φ j. We are able to decompose a fuctio φ ito a sum of r fuctios which is the umber of cojugacy classes of G. 2. Mai Result To state the mai results we first defie ad ivoke few results take from [5]. R ψ,φ (τ = α i (tα j (tτ. Lemma 1 Let ψ, φ be the Fourier trasforms of ψ ad φ respectively. The we have Lemma 2 Let φ l 2 (G. The R ψ,φ = ψ φ. if ad oly if R φ,φ = φ φ = φ(ρ 1 φ(ρ 2 φ(ρ r with φ(ρ j = Q j, where Q j is a projectio matrix for all j {1, 2,..., r}.

5 ON THE EXISTENCE OF A GROUP ORTHONORMAL BASIS 49 Theorem 3 Let φ l 2 (G ad let {ρ j } be the set of all irreducible represetatios of G, ρ j has degree d j. Let ρ j (k, l be the (k, l etry i the d j d j matrix of ρ j. Cosider the (ivoluted fuctio ρ j (k, l(g 1 l 2 (G. The the (diagoal set of fuctios { } dj ρ j(k, k is a G-orthoormal set. j=1,...,r; k=1,...,d j Note the set of fuctios { } dj ρ j(k, l j=1,...,r; k,l=1,...,d j is ot G-orthoormal. Thus we have geerated m G-orthoormal basis vectors for l 2 (G, where m = r j=1 Now we are ready to state our mai result of the paper. Theorem Let G be a group of elemets. The l 2 (G admits a G-orthoormal basis if ad oly if G is a abelia group, i which case this basis is uique up to a multiplicatio by a complex umber of modulus oe. Proof: The case of G beig cyclic is trivial ad the complex expoetials yield (up multiplicatio the uique G-orthoormal basis for l 2 (G due to Lemma 2 ad the observatio that the Fourier coefficiets are just complex umbers. The case of a abelia group follows readily as ay abelia group is a direct sum of cyclic groups. Now assume G i o-abelia. The the group G has r cojugacy classes where r < ad therefore there is at least oe irreducible group represetatio of degree greater tha oe. The ecessary coditio to have a G -orthoormal basis with elemets i l 2 (G is to produce d 2 j mutually orthoormal projectios matrices i M dj d j (C. This is, however, impossible ad the maximum umber of mutually orthoormal projectio matrices i M dj d j (C is equal to d j. Suppose {Q i } is a set of m mutually orthoormal projectio matrices i M dj d j (C. I particular, we have Q i Q j = Q i δ i,j for all i, j {1, 2,..., m}. It is readily see that the sum d j m Im(Q i i=1 must be direct as Im(Q j Ker(Q i for all i, j {1, 2,..., m} ad i j. The result the follows. Corollary Let G be a o-abelia group of elemets. The the (diagoal set of fuctios {ρ j (k, k} j=1,...,r; k=1,...,dj is G-orthoormal set of m fuctios i l 2 (G, where m = r j=1 d j. Ay other G-orthoormal set i l 2 (G has the umber of elemets less tha m.

6 50 PETER ZIZLER If we relax the coditio for G-orthoormality to just G decorrelatio, i particular α i (tα j (tτ = 0 for all τ G ad i j the such a (G decorrelated basis exists, a example of such is the set of fuctios {ρ j (k, l} k,l {1,2,...,dj}. I the cyclic case the complex expoetials are eigefuctios to a covolutio operator C ψ for ay ψ l 2 (G. I the o-abelia case this is o loger true i geeral. However, the followig subspace, of dimesio d j, is a ivariat subspace to a covolutio operator C ψ for ay ψ l 2 (G, M j,k = spa{ρ j (k, l l {1,... d j }}. It turs out that ay subspace that is a ivariat subspace to ay covolutio operator C ψ for ay ψ l 2 (G, must have a dimesio of at least d j, see [6]. A fuctio φ : l 2 (G C is called a multiplicative character if φ(στ = φ(σφ(τ for all σ, τ G. Observe that multiplicative characters are the etry fuctios ρ j (1, 1 with d j = 1 ad thus are clearly are eigefuctios to a covolutio operator C ψ for ay ψ l 2 (G. We say that a set {α i } m i=1 is a maximal G-orthoormal set i l2 (G if α i (tα j (tτ = α i (τδ i,j for all τ G ad m = r j=1 d j, where {d j } r j=1 are the degrees of irreducible represetatios of G. Cosider a G-orthoormal set {α i } m i=1 i l2 (G. Let {i 1,..., i k } ad {j 1,..., j l } be two disjoit subsets of {1, 2,..., m}. The we ca defie β i = k c i α is ad β j = s=1 l c j α is ad observe that {β i, β j } forms a G-orthoormal set i l 2 (G if the modulus of the complex umbers c i ad c j is oe. From a give G-orthoormal set we ca costruct other G-orthoormal sets, typically with fewer elemets. The motivatio behid this, typically, is to obtai real valued G-orthoormal fuctios. I the cyclic case we ca form e 2πijk + e 2πijk ( 2πjk = 2 cos or ie 2πijk s=1 ad i the o-abelia case we ca form (for g G + ie 2πijk ( 2πjk = 2 si d j ρ j (k, k(g 1 = tr(ρ j (g k=1

7 ON THE EXISTENCE OF A GROUP ORTHONORMAL BASIS 51 ad we get the irreducible characters which form a G-orthoormal set i l 2 (G of size r, the umber of cojugacy classes of G. 3. The Space l 2 (S 3 We will cosider the symmetric group S 3 i our example. The group G = S 3 cosists of elemets g 0 = (1 ; g 1 = (12 ; g 2 = (13 ; g 3 = (23 ; g 4 = (123 ; g 5 = (132. The group S 3 has three cojugacy classes {g 0 }, {g 1, g 2, g 3 }, {g 4, g 5 }. We have three irreducible represetatios, two of which are oe dimesioal, ρ 1 is the idetity map, ρ 2 is the map that assigs the value of 1 if the permutatio is eve ad the value of 1 if the permutatio is odd. Fially, we have ρ 3, the two dimesioal irreducible represetatio of S 3, defied by the followig assigmet ( g 3 g 0 ( e 2πi/3 e 2πi/3 0 ( ( 0 1 ; g 1 ; g ( e 2πi/3 0 ; g 4 0 e 2πi/3 The irreducible characters of S 3 are give by χ 1 = (1, 1, 1, 1, 1, 1 T χ 2 = (1, 1, 1, 1, 1, 1 T φ 3 = (2, 0, 0, 0, 1, 1 T 0 e 2πi/3 e 2πi/3 0 ; g 5 where χ 1 ad χ 2 are also multiplicative characters. Moreover, we have ρ 3 (1, 1 = (1, 0, 0, 0, e 2πi/3, e 2πi/3 T ρ 3 (1, 2 = (0, 1, e 2πi/3, e 2πi/3, 0, 0 T ρ 3 (2, 1 = (0, 1, e 2πi/3, e 2πi/3, 0, 0 T ρ 3 (2, 2 = (1, 0, 0, 0, e 2πi/3, e 2πi/3 T. ( e 2πi/3 0 0 e 2πi/3. We have m = 4 ad Note that the set of fuctios { } χ 1, χ 2, ρ 3 (1, 1, ρ 3 (2, is a maximal G-orthoormal set i l 2 (S 3. However, the set { } χ 1, χ 2, ρ 3 (1, 1, ρ 3 (1, 2, ρ 3 (2, 1, ρ 3 (2, forms a G-decorrelated basis for l 2 (S 3.

8 52 PETER ZIZLER Refereces [1] D. S. Dummit ad R. N. Foote, Abstract Algebra, Joh Wiley ad Sos, [2] G. James ad M. Liebeck, Represetatios ad Characters of Groups, Cambridge Uiversity Press, [3] J.-P. Serre, Liear Represetatios of Fiite Groups, Spriger, [4] R. S. Stakovic, C. Moraga ad J. T. Astola, Fourier Aalysis o Fiite Groups with Applicatios i Sigal Processig ad System Desig, IEEE Press, Joh Wiley ad Sos, [5] P. Zizler, O spectral properties of group circulat matrices, PaAmerica Mathematical Joural, 23 (1 (2013, pp [6] P. Zizler, O the amplitude ad phase respose i the o-abelia Fourier trasform, preprit. Peter Zizler Departmet of Mathematics, Physics ad Egieerig Mout Royal Uiversity, Calgary, Caada pzizler@mtroyal.ca