ON KAC PARAMETERS AND SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS OF LIE ALGEBRA sl n. Zlatko Drmač and Tomislav Šikić

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1 RAD HAZU MATEMATIČKE ZNANOSTI Vo 18 = 519 (2014): ON KAC PARAMETERS AND SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS OF LIE ALGEBRA s n Zatko Drmač and Tomisav Šikić Abstract This paper presents interesting spectra properties of a particuar integer skew-symmetric matrix, used to encode information on Z-gradation of type s for cassica affine Lie agebra sn It is shown that the hidden Kac parameters can be reveaed using an expicity computed eigenvector in a Gram-Schmidt orthogonaization process 1 Introduction The connection between the Z-gradation of type s of an affine Lie agebra g and the finite order automorphisms of the simpe Lie agebra g is among the eariest important achievements of the theory of affine Lie agebras (see [7], [8] or [5]) In 1968, V G Kac ([7]) showed that a N-th order automorphisms of g are, in fact, parameterized (in such a way) by a sequence of nonnegative reativey prime integers (s 0, s 1,, s ) for which N = k i=0 a is i Here the a i s are positive integer abes of the corresponding Dynkin Diagram X (see [8]) The sequence (s 0, s 1,, s ) is in [14] designated by the term Kac parameters and denoted by s Kac Further, [14] provides an agorithm to determine s Kac for every Z-gradation of the type s for the cassica{ affine Lie agebras, ie, } Lie agebras with Dynkin diagrams of the type X A (1), B (1), C (1), D (1) An increased interest for researching various gradations, corresponding speciaizations of Wey-Kac character formua for affine Lie agebras and for certain reaizations of modues (ie representations) by vertex operators is triggered in the eary 1980 s, et us just mention [13], [9], [10] Many of the vertex operator constructions of integrabe highest weight representations and the corresponding Z-gradations and speciaizations are not described in terms of Kac parameters, see eg [10], [12], [11] 2010 Mathematics Subject Cassification 17B67, 15A18, 15B57 Key words and phrases Z-gradation for cassica (affine) Lie agebra, Kac parameters of finite order automorphisms of simpe Lie agebras, matrix of speciaized roots, integer skew-symmetric matrix, spectra decomposition 55

2 56 Z DRMAČ AND T ŠIKIĆ The reaization of the cassica simpe Lie agebra by traceess matrices (see [3]) enabes us to find the connection between their Z-gradations of type s = (s 1,, s n 1 ) and certain n n integer skew-symmetric matrix, caed the matrix of speciaized roots, and denoted by MSR X (s) In fact, this matrix is a convenient encoding of the corresponding Z-gradation of the type s of the cassica simpe Lie agebra (see Definition 21) Using some basic properties of the matrix of speciaized roots, the agorithm [14] extracts the proper eement from the Wey group which points to hidden Kac parameters In this process of finding the Kac parameters, the agorithm rearranges a eements of MSR X (s) in an interesting way Especiay, in the case of Lie agebra with X = A (1), the upper triange becomes nonnegative, and the ower triange becomes nonpositive This interesting phenomenon motivated the authors of this paper to study the structure of the matrix of speciaized roots A = MSR (1) A (s) Athough integer, skew-symmetric and of rank ony two, A possesses very intriguing structure For instance, an eigenvector corresponding to a nonzero eigenvaue determines a eements of the Wey group that point to the hidden Kac parameters simpe sorting of the components of the imaginary part of this eigenvector reveas the key permutation(s) Furthermore, such an eigenvector of a nonzero eigenvaue can be computed by an expicit formua its rea part is constant and the imaginary part is obtained by the Gram- Schmidt orthogonaization of the vector of partia sums of s= (s 1,, s n 1 ) against the vector of ones Thus, using pure inear agebra, we have devised a simpe and efficient agorithm for computing the key eements of the Wey group of s n As an iustration of its power, we easiy reproduce the resuts of the agorithm in [14], see Coroary 38 The rest of the paper is organized as foows In 2, we set the stage, define the matrix of speciaized roots and reca the resuts from [14] In 3 we first observe that the matrix of speciaized roots is of rank two, and give expicit formuas for its eigenvaues and eigenvectors We show that the permutation that sorts the imaginary part of an eigenvectors reveas the underying structure Section 4 presents an appication of our resuts to bosonic and fermionic reaization [11], which is connected with some Z-gradations of type s 2 The matrix of speciaized roots of s n It is we known that traceess n n matrices form the Lie subagebra g = s n of the genera inear (Lie) agebra g n Ceary, dim(s n ) = n 2 1, with the standard basis B = { E ij e i e T j i j} {H i E ii E i+1,i+1 i = 1,, n 1} (Here e i denotes the i-th coumn of the n n identity matrix I, and E ii = e i e T i )

3 SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS 57 Now it is very important to notice that the set {H i i = 1,, n 1} spans one maxima tora subagebra h (consisting of semisimpe eements) In the case when g is (semi)simpe, the Lie subagebra h is Cartan subagebra Since h is abeian, the set ad g h = {ad g h h h} is simutaneousy diagonaizabe, where ad g h(x) = [h, x], x g Hence, g is the direct sum of the subspaces g α = {x g ad(h)(x) = α(h)x, h h}, where α is a inear functiona (α h ) The set of a nonzero α h for which g α 0 is usuay denoted by R and caed the root system The eements from R are caed the roots (of g reative to h) Foowing this notation, we have (21) g = g 0 + α Rg α, where g 0 = h The decomposition (21) is caed a Cartan or root space decomposition In the case of s n, the root system R corresponds to the Dynkin diagram A n 1 and = (α 1,, α n 1 ) denotes the standard basis of the root system R, ie each root α R can be written uniquey as n 1 α = k i α i, i=1 with integra coefficients k i a nonegative or a nonpositive Let {ˆε i, i = 1,, n} be a dua basis of the basis {E ii, i = 1,, n} of the diagona matrices from g n Let ε i be the restriction of ˆε i on the Cartan subagebra h It is we known (see [2] and [3]) that R = {±(ε i ε j ), 1 i < j n} (22) α 1 = ε 1 ε 2, α 2 = ε 2 ε 3,, α n 2 = ε n 2 ε n 1, α n 1 = ε n 1 ε n Let now s = (s 1,, s n 1 ) be an (n 1)-tupe of integers and α = n 1 i=1 k iα i an arbitrary root Define the mapping deg : R Z by n 1 (23) deg(α) = k i s i i=1 For the roots from the basis, it hods that (24) deg(α i ) = s i i = 1,, n 1 By (22) we can interpret (24) as a system of inear equations with variabes deg(ε i ) (25) deg(ε i ) deg(ε i+1 ) = s i i = 1,, n 1 We reca the foowing notion of the matrix of speciaized roots [14]

4 58 Z DRMAČ AND T ŠIKIĆ Definition 21 Let = (α 1,, α n 1 ) be a basis of the simpe Lie agebra root system R (for simpe Lie agebra s n ) Let s = (s 1,, s n 1 ) be an arbitrary (n 1)-tupe of integers, and et the mapping deg : R Z for the basis be given by (23) The matrix [a ij ] i,j Z n n, defined by (26) a ij = deg(ε i ) deg(ε j ) i, j {1,, n} is caed the matrix of speciaized roots and denoted by MSR (1) A (s) n 1 Remark 22 Note that the vaues of variabes deg(ε i ) are not uniquey determined by (25), but the above definition does not depend on the choice of soutions deg(ε i ) Indeed the functiona ε i ε j for i j is aways a root and we have foowing equations a i,j def n 1 = deg(ε i ) deg(ε j ) = deg(ε i ε j ) = deg(α) = deg( k i α i ), where α = ε i ε j For i = j it is evidenty a i,i = deg(ε i ) deg(ε i ) = 0 Remark 23 In [14], the term matrix of speciaized roots is simutaneousy defined for a cassica simpe Lie agebras, ie for Lie agebras with Dynkin diagrams of the type X {A (1), B (1), C (1), D (1) }, and the corresponding notation is MSR X (s) For the sake of brevity, in this paper we focus ony on the case s n (ie Dynkin diagram A (1) n 1 ) and we simpify the notation by writing MSR(s) MSR (1) A (s) Our preiminary resuts indicate that simiar n 1 deveopment is possibe for other Dynkin diagrams Exampe 24 Let g = s 5 and s = (s 1, s 2, s 3, s 4 ) = (3, 11, 5, 2) The root system is R = {±α 1, ±α 2, ±α 3, ±α 4, ±(α 1 + α 2 ), ±(α 2 + α 3 ), ±(α 3 + α 4 ), i=1 ±(α 1 + α 2 + α 3 ), ±(α 2 + α 3 + α 4 ), ±(α 1 + α 2 + α 3 + α 4 )} The matrix of speciaized roots reads MSR(s) = It is important to notice that every deg(α) for α R is paced on (i, j)- position in the matrix which reated to corresponding root vector For instance, deg(α 2 + α 3 )(= 16) is paced on the (2, 4)-position At the same time the matrix E 2,4 is root vector for the root α 2 + α 3 h, ie [h, E 2,4 ] = (α 2 + α 3 )(h) E 2,4, h h

5 SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS MSR(s) and the Wey group As we emphasized in the introduction, the matrix of speciaized roots MSR X (s) has one of the main roes in the agorithm for finding Kac parameters of Z-gradation of type s for cassica affine Lie agebras In fact, the matrix MSR(s) wi serve to detect another base of the root system R that wi guarantee the positivity of the Kac parameters (see [14]) Before we show our proposition about the connection between matrix MSR(s) and the hidden base we introduce an additiona notation reated to the Wey group Let E be the ambient Eucidean space for the root system R The subgroup of GL(E) generated by the refections σ α (β) = β 2(β, α) (α, α) α, α R is the Wey group W Since any refection σ α eaves the roots system R invariant, the Wey group in fact permutes the set R Moreover W acts simpy transitivey on bases, ie i) if is another base of R, then = σ( ) for some σ W ; ii) if = σ( ) then σ = id Since Wey group of s n is isomorphic to symmetric group S n, its cardinaity rapidy grows with increasing n, W = n! Hence, to determine an appropriate base of the root system R, the naive search is not feasibe The foowing proposition gives us the properties of MSR(s), which wi enabe us to find an appropriate base via corresponding eement from Wey group Proposition 25 Let MSR(s) be the matrix of speciaized roots for an arbitrary (n 1)-tupe of integers s = (s 1,, s n 1 ) Then 1 The MSR(s) is an skew-symmetric matrix 2 The coumns Ae i, i = 1,, n, of A can be ordered so that Ae j1 Ae j2 Ae jn 1 Ae jn, where the inequaity between two vectors is understood entry-wise 3 The matrix MSR(s) contains at east one row composed of nonnegative integers 4 The permutation π = ( 1 2 n 1 n ) j 1 j 2 j n 1 j n Sn ( W) yieds the basis = (π(α 1 ),, π(α n 1 )) of the root system R, which satisfies the required condition deg(π(α i )) = a ji,j i+1 Z +, i = 1,, n 1 5 For each cycic permutation (i 1, i 2,, i r 1, i r ) S n, it hods that a i1,i 2 + a i2,i a ir 1,i r + a ir,i 1 = 0

6 60 Z DRMAČ AND T ŠIKIĆ Proof (1) From the definition (26) it foows that MSR(s) is skewsymmetric (2) Let i, j, k be arbitrary indices from {1, 2,, n} Using (26), we have (27) a ij a ik = deg(ε i ) deg(ε j ) deg(ε i ) + deg(ε k ) = deg(ε k ) deg(ε j ) = a kj Hence, the difference between two eements does not depend on the choice of the row index In fact, we can simutaneousy order eements for a rows in the foowing way (28) a i,j1 a i,j2 a i,jn 1 a i,jn, i = 1,, n This impies that the caim (2) is true (3) Since a i,i = 0, (28) impies that the j1 th row consist of positive eements (4) It is we known that permutation π is an eement of the Wey group of Lie agebra s n and the Wey group W of any simpe Lie agebra acts transitivey on bases (see for instance [6]) Since the mapping between the roots α i π(α i ) is determined by mapping between the functionas ε k ε jk, k = 1, 2,, n, we can write deg(π(α i )) = deg(π(ε i ε i+1 )) = deg(ε ji ε ji+1 )) = a ji,j i+1 for every i = 1, 2,, n 1 Using (27) and (28), we immediatey concude that deg(π(α i )) = a ji,j i+1 = a i,ji+1 a i,ji 0 (5) Let (i 1, i 2,, i r ) be an arbitrary cycic permutation and denote a cyce = a i1,i a ir 1,i r + a ir,i 1 By definition of MSR(s) (26) a cyce = deg(ε i1 ε i2 ) + + deg(ε ir 1 ε ir ) + deg(ε ir ε i1 ) = deg[(ε i1 ε i2 ) + + (ε ir 1 ε ir ) + (ε ir ε i1 )] = deg(0) = 0 Remark 26 From Proposition 25 ( 5) we know that for any cosed circuit of any ength k 2, a i1i 2 + a i2i a ik 1 i k + a ik i 1 = 0 Thus, the tropica spectra radius of A is 22 An exampe ρ trop (A) = max i 1,,i k a i1i 2 + a i2i a ik 1i k + a ik i 1 k Exampe 27 Take n = 12 and = 0 s = ( )

7 SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS 61 The matrix MSR(s) is given by MSR(s) = The 10 th row of MSR(s) contains ony nonnegative integers (a 10,j) = We can order the seected row a 10,10 a 10,6 a 10,9 a 10,3 a 10,1 a 10,8 a 10,2 a 10,5 a 10,4 a 10,7 a 10,11 a 10, This determines the proper eement of the Wey group ( π = which points to the right base, and to the corresponding (n 1)-tupe of nonnegative integers ), s = (a 10,6, a 6,9, a 9,3, a 3,1, a 1,8 a 8,2, a 2,5, a 5,4, a 4,7, a 7,11, a 11,12 ) = Then the functionas = (4, 1, 1, 1, 0, 2, 0, 1, 0, 0, 11) α 1 = ε 10 ε 6 = (α 6 + α 7 + α 8 + α 9 ) α 2 = ε 6 ε 9 = α 6 + α 7 + α 8 α 3 = ε 9 ε 3 = (α 3 + α α 8 ) α 4 = ε 3 ε 1 = (α 1 + α 2 + α 3 ) α 5 = ε 1 ε 8 = α 1 + α α 7 α 6 = ε 8 ε 2 = (α α 7 ) α 7 = ε 2 ε 5 = α 2 + α 3 + α 4 α 8 = ε 5 ε 4 = α 4 α 9 = ε 4 ε 7 = α 4 + α 5 + α 6 α 10 = ε 7 ε 11 = α α 10 α 11 = ε 11 ε 12 = α 11 form the basis = (α 1,, α n 1) The matrix of speciaized roots MSR(s ) is MSR(s ) = Note that

8 62 Z DRMAČ AND T ŠIKIĆ The permutation π redistributes the signs so that in the resuting skew symmetric matrix the upper triange is positive, and the ower negative The entries in each coumn and in each row are monotonicay ordered The difference of any two rows (or any two coumns) of MSR(s ) is a vector with a entries equa The same property hods for MSR(s) (See Proposition 25) 3 Spectra decomposition of MSR(s) The matrix of speciaized roots can be written in compact form as a difference of two dyads (rank-one matrices), parametrized by the integers s i This eads to a different and more natura parametrization and aows detaied spectra anaysis of MSR(s) as a function of s, without using the matrix entries (26) 31 A dyadic representation of MSR(s) Our key observation is the foowing simpe proposition Proposition 31 Let s = (s 1,, s n 1 ) be an arbitrary (n 1) tupe of integers The matrix of speciaized roots MSR(s) can be represented as (31) MSR(s) A = EST T T S T E T, where E = ( ) T ( ), S = 0 s1 s n 1, T = The vector S = (0 s) is uniquey determined by A: S = e T 1 T T AT 1, s = e T 1 T T AT 1 (e 2 e n ) Further, it hods E T AE = SAS T = 0 Proof From the Definition 21, it immediatey foows a i,i+1 = s i, i = 1,, n 1 (cf (25)), and from [14] we know that a i,j = a i,i+1 + a i+1,i a j 2,j 1 + a j 1,j Hence, for i < j, a ij = s i + + s j 1 Since A is skew symmetric by definition, a ji = a ij On the other hand, it is straightforward to check that the righthand side of (31) gives precisey the same eements of A The remaining caims are easiy proved; the ony sma technica detais is to note that the inverse of T is an upper bidiagona matrix with unit diagona and the vaue of minus one at the first superdiagona Remark 32 Note that we have actuay defined a inear mapping M( ) : R n 1 Skew n, M(s) = EST T T S T E T into the vector space Skew n of skew symmetric matrices It is easy to show that M(s) = 0 n n if and ony if s = 0 n 1 It is aso cear that the matrices MSR(s) are integer vectors in an (n 1) dimensiona inear subspace of Skew n

9 SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS Spectra decomposition The Schur form of A is diagona (see eg [1]), and we want expicit formuas for its eigenvaues and eigenvectors The specia structure of A certainy raises expectations regarding the structure of its spectra decomposition Our goa is not to merey compute the eigenvaues and the eigenvectors, but to discover more of the intrinsic structure of A that may, in turn, impart new knowedge in the Lie agebra setting In the seque, the Eucidean inner product is denoted by <, >, and = <, > is the corresponding induced norm The imaginary unit is i = 1 Theorem 33 Let s = (s 1,, s n 1 ) be an arbitrary (n 1) tupe of integer scaars, 1 and et A = EST T T S T E T, where E, S, T are as in Proposition 31 Further, set and et P = (ST ) T = ( 0, s 1, s 1 + s 2,, s s n 1 ) T, 1 (32) P = P E T E EET P be the resut of the Gram Schmidt orthogonaization of P to E Then A EP T P E T has spectra decomposition A = W ΛW, where Λ = diag(λ 1,, λ n ), W W = I n, and λ 1 = i <P, P><E, E> <P, E> 2, λ 2 = i <P, P><E, E> <P, E> 2, λ 3 = λ 4 = = λ n = 0 The corresponding eigenvectors (the coumns of W ) are w 1 = w 2 = w 1 w 1, where w 1 = P E E +i P, w 2 w 2, where w 2 = P E E i P, w 3,, w n = any orthonorma basis of span(w 1, w 2 ) Proof We first note that A 0 is of rank two There are many simpe ways to show ( this For instance, the congruence B T AB with B = T 1 yieds B T 0 s AB = s T 0 n 1,n 1 ), which is of rank two for any s 0 (Note here that B is upper bidiagona matrix with unit diagona and with the vaue 1 on the first super diagona In fact, B is the Choesky factor of Cartan matrix) The nu space of A is easiy cacuated as Ker(A) = span(e, P ) 1 At this point, we are ony interested in integer s i s; going over to rea or compex s foows mutatis mutandis

10 64 Z DRMAČ AND T ŠIKIĆ Hence, Im(A) = span(e, P ) If we orthogonaize P against E to compute P = P 1 n EET P, then span(e, P ) = span(e, P ), E T P = 0 Hence, A has ony two non zero eigenvaues of the form (due to skew symmetry) λ 1 = iω, λ 2 = iω, where ω R + The remaining eigenvaues are λ 3 = = λ n = 0 It is easiy checked that the negative semidefinite A 2 = A T A has doube nonzero eigenvaue with eigenvectors E and P : ( A 2 n E = ζe n 1 A 2 P = ζp with ζ = k s k (n k)) n s j < 0 k=1 Note that ζ = (E T P ) 2 np T P, that is, ζ =<P, E> 2 <P, P ><E, E> Since P and E are not coinear, ζ < 0 by the Cauchy Schwarz inequaity The number ζ is doube eigenvaue of A 2, and the corresponding eigenspace is span(e, P ) = span(e, P) We immediatey concude that ω = ζ Let w 1 = x + iy be an (essentiay unique) eigenvector beonging to λ 1 Then w 2 = x iy is an eigenvector beonging to λ 2 Further, we equivaenty write Ax = ωy, Ay = ωx, ie A ( x y ) = ( x y ) ( ) 0 ω ω 0 From x T Ax = ωx T y it foows x T y = 0, ie x and y are perpendicuar in the Eucidean inner product Comparing y T Ax and x T Ay we concude that aso x T x = y T y, ie x and y are of the same Eucidean ength If we take x 2 = y 2 = 1, then A = ω(xy T yx T ) It remains to find x and y Note that, due to skew symmetry, k=1 j=1 A P = η 1 E, η 1 = ET A P E T E = ET AP E T E, and AE = η 2 P, η2 = ET AP P T P Putting the two reations together and re scaing yieds ( A P E E Hence, we can write P ) ( = P E E ω = ET AP E P, λ 1 = iω, P ) 0 E T AP E P 0 ET AP E P w 1 = P E E +i P Remark 34 Note that the proof contains the identity ζ = ω 2 = η 1 η 2 =<P, P><E, E> <P, E> 2 = <AP, E> 2 <E, E>< P, P>,

11 SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS 65 and that we can say that the spectrum of A is in ±i Z + Further, if we compare the first entries in the reation AE = η 2 P, using the representation A = EP T P E T, we immediatey get η 2 = n Aso, A P = η 1 E yieds η 1 = P T P Remark 35 An eigenvector w 1 = x +iy is determined up to mutipication with non zero compex scaar α +iβ Take an arbitrary α +iβ 0 and consider (α +iβ)(x +iy) = x +iỹ Note that (33) ) ( ) ( x ( α/ρ β/ρ ỹ = ρ x y β/ρ α/ρ ), ρ = α +iβ If the scaing factor is such that α + iβ = 1, then this is just a change of orthogona basis in the image of A Coroary 36 For an arbitrary choice of the eigenvector w = x + iy of the eigenvaue λ 1 (or λ 2 ), the n points P j = (x j, y j ), j = 1,, n, are coinear Proof The caim obviousy hods true for the particuary chosen eigenvectors w 1 and w 2 from the Theorem 33 Using Remark 35 and the fact that transformation (33) preserves coinearity 2 of the points (given by two coordinates in each row of ( x y ) ) the caim remains true for any other eigenvector of λ 1 (λ 2 ) Coroary 37 Let π be a permutation that sorts the components of the vector P in a non-increasing sequence, and et w = x + iy be an arbitrary eigenvector beonging to λ 1 or λ 2 Then π is a sorting permutation for both x and y Proof Again, the caim obviousy hods true for the particuary chosen eigenvectors w 1 and w 2 from the Theorem 33 Next note that, due to coinearity, a permutation that sorts the components of x, aso sorts the components of y (but not necessariy in the same direction, eg y may get sorted in non-decreasing sequence, whie x is sorted in non increasing sequence) And finay, since any transformation from Remark 35, with ρ = 1, just rotates the ine with the points as a rigid body, the order is preserved, up to the direction An additiona scaing with ρ 1 preserves the order Coroary 38 Let Π be the matrix representation of the permutation π from Coroary 37 Then π is aso a sorting permutation for P Further, a entries in the upper triange of Π T AΠ are non negative; by skew symmetry a entries in the ower triange are non positive In each row of Π T AΠ the entries are monotonicay increasing; in each of its coumns, the entries are monotonicay decreasing For each custer of k equa entries in Π T P, the 2 Use the determinant criterion for any three points

12 66 Z DRMAČ AND T ŠIKIĆ matrix Π T AΠ has a k k zero bock on the main diagona (rows) with indices in such a zero bock are mutuay equa A coumns as Proof We use the formuas from the proof of Theorem 33 to write A A EP T P E T = ω P 2 E 2 (E P T P E T ) The concusion about the distribution of the signs in Π T AΠ foows from the foowing simpe fact: if the entries of a vector v are non decreasing, v 1 v n, then (Ev T ve T ) ij 0 for 1 i j n, and in each row (coumn) of Ev T ve T the entries are monotonicay increasing (decreasing) Coroary 39 The exponentia of A can be expicity written as the rotation e A = I + sin ω ω A + 1 cos ω ω 2 A 2, ω = A 2 Proof It is quite easy to setup the quadratic Lagrange poynomia that resembes the exponentia function on the spectrum of A: L 2 (iω) = e iω, L 2 ( iω) = e iω, L 2 (0) = 1 Hence, L 2 (z) = 1 + (sin ω/ω)z + ((1 cos ω)/ω 2 )z 2, and e A = L 2 (A) Note that this is an instance of the Rodrigues formua for 3 3 rotation, and it aso foows as a specia case from the generaized Rodrigues formua [4], since it hods A 3 = ω 2 A Exampe 310 (Exampe 27 continued) The vector P reads P = 1 ( ) T Its sorting permutation is the same π as in Exampe 27, and the permuted matrix is Π T AΠ = MSR(s ), as in Exampe 27 Atogether 3! 2! 2! permutations eave the structure of Π T AΠ intact This ambiguity of the sorting permutation is easiy seen in the repeated entries of the vector P 4 Spectra decomposition of MSR(s) in the case of bosonic and fermionic reaization As we pointed out in 1, many of the vertex operator constructions of integrabe highest weight representations and the corresponding gradations and speciaizations are not described in terms of Kac parameters For instance, in [11] the bosonic and fermionic reaization of the affine agebra g n for the conjugacy casses of Heisenberg subagebra are parametrized by partitions of the integer n n = {n 1, n 2,, n r }; n 1 n 2 n r

13 SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS 67 The associated Z-gradations of the constructed modues are determined by n-tupe s = (s 0,, s n 1 ) of reativey prime integers defined by ( n1 + n r 1 s = N,,, 1, n 1 + n 2 1, 1 1,,, n 2 + n 3 1, 2n 1 n r n 1 n }{{ 1 2n } 1 n 2 n 2 n }{{ 2 2n } 2 n 3 n 1 1 n ,, n ) r 1 + n r 1 (41),, n r 1 n r 1 }{{} n r 1 1 1,,, 1 2n r 1 n r n r n r }{{} n r 1 In this particuar construction, the positive integer N is computed as { N, if N ( 1 n i + 1 n j ) 2Z for a i, j {1,, r}; (42) N = 2N if N ( 1 n i + 1 n j ) / 2Z for some (i, j), where N is the east common mutipe of {n 1, n 2,, n r } Since N( ni+ni+1 2n in i+1 1) are negative integers, the above Z-gradations are not parametrized by Kac parameters The corresponding (n 1)-tupe, derived from (41) reads s = ( s 1 s 2 s n 1 ) (43) = ( N,, N, N n 1 + n 2 N, N,, N, N n 2 + n 3 N, n 1 n }{{ 1 2n } 1 n 2 n 2 n }{{ 2 2n } 2 n 3 n 1 1 n 2 1 N N,,,, N n r 1 + n r N, N,, N ) n r 1 n r 1 2n r 1 n r n r n r }{{}}{{} n r 1 1 n r 1 Exampe 41 (See [14, Exampe 38]) Take n = 12 and {n 1, n 2, n 3 } = {3, 4, 5} For a given partition we have N = 120 and s = ( 40, 40, 85, 30, 30, 30, 93, 24, 24, 24, 24 ) The corresponding matrix MSR(s) has the form MSR(s) = Foowing the agorithm from [14] we have the 8 th row of nonnegative integers (a 8,j) =

14 68 Z DRMAČ AND T ŠIKIĆ We can order the seected row and determine the eement of the Wey group ( ) π =, which points to (n 1)-tupe of nonnegative integers s = (a 8,4, a 4,1, a 1,9, a 9,5, a 5,2, a 2,10, a 10,6, a 6,11, a 11,3, a 3,7, a 7,12 ) = (3, 5, 16, 9, 15, 0, 15, 9, 16, 5, 3) Since N = 120, the associated Z-gradation is determined by reativey prime integers s = ( 32, 40, 40, 85, 30, 30, 30, 93, 24, 24, 24, 24 ) The hidden Kac parameters for the mentioned bosonic and fermionic reaization are s Kac = (s Kac 0,, s Kac 11 ) = (24, 3, 5, 16, 9, 15, 0, 15, 9, 16, 5, 3) where s Kac 0 = a 12,8 + N = = 24 Of course, thanks to Theorem 33 (or more precisey Coroary 38) the corresponding permutation Π redistributes the signs so that in the resuting skew-symmetric matrix the upper triange is positive, and the ower negative Π T AΠ = Note that the Kac parameters are paced on the first superdiagona in the matrix Π T AΠ The specia structure (43) of s impies additiona structure of the key vector P defined in (32) Theorem 42 Let n = {n 1,, n r } be a partition of n Let s be given by (43) Then P 1 2 Zn and its nonzero entries come in ± pairs (hence, for even (odd) n an even (odd ) number of entries of P can be zero) More precisey, the sign structure of P can be described as foows: First, partition P as ) ( P (44) P = [1] P [r], P [i] 1 2 Zni, i = 1,, r

15 SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS 69 Then the vectors P [i] have the foowing sign structure: (45) (46) if n i is even: P [i] j if n i is odd: { P [i] j [i] = P n < 0, ; j = 1,, n i i j+1 2 [i] ni 1 = P n i j+1 < 0, ; j = 1,, 2 P [i] n i = 0 Proof We first show that P 1 2 Zn Since P is integer vector and E T E = n, it suffices to show that 1 n ET P 1 2 Z, ie 1 n EET P 1 2 Zn Note that (47) 1 n ET P = s n (s 1 + s 2 ) + + (s 1 + s s n 1 ) = (1/n) s k (n k) n Set M = n 1 n r By induction with respect to the index r, it can be shown that n 1 (48) M ṡ k (n k) = 1 2 (n 1 1)n 2 n r n, where ṡ k = s k N k=1 Then we can write (47) as 1 n ET P = (1/n) N n 1 Mṡ k (n k) = (1/n) N M M 1 2 (n 1 1)n 2 n r n (49) k=1 = N M 1 2 (n 1 1)n 2 n r = N 2n 1 (n 1 1) From N n 1 Z and (n1 1) Z we can concude that 1 n ET P 1 2 Z, ie P 1 2 Zn Since P and E are orthogona, it must hod n P i=1 i = 0; so we know that P must have entries of both signs Since the numbers s i are particuary structured, some additiona structure of P is expected For instance, the first bock in the partition (44) reads P [1] = 0 s 1 s 1+s 2 s 1+s 2+ +s n1 1 N(n 1 1)/(2n 1) N(n 1 1)/(2n 1) N(n 1 1)/(2n 1) N(n 1 1)/(2n 1) k=1, where s 1 = = s n1 1 = N, n 1 and one can easiy check that the entries of P [1] are monotonicay increasing and that P [1] j + P [1] n 1 j+1 = (j 1) N n 1 N(n 1 1) 2n 1 + (n 1 j) N n 1 N(n 1 1) 2n 1 = 0

16 70 Z DRMAČ AND T ŠIKIĆ The structure of the second bock is sighty more compicated and we have P [2] [2] j + P n = s 2 j s n1 + (j 1) N N(n 1 1) + s s n1 n 2 2n 1 + (n 2 j) N n 2 N(n 1 1) 2n 1 = 0, where we have used that s s n1 = N( n 1 1 n 1 + n 1 + n 2 2n 1 n 2 1), s n1+1 = = s n1+n 2 1 = N n 2 The genera case is anaogous but sighty more technica, and we skip the detais for the sake o brevity Coroary 43 Let the assumptions of Theorem 42 hod true If, in addition, either N = 2N or n = {n 1,, n r } is a partition of n with reativey prime numbers n i, then P is an integer vector Proof Using the cacuation (49) we have the foowing discussion Suppose that N = 2N From N 2n 1 = N n 1 it is obvious that 1 n ET P Z, ie P is an integer vector Now et n = {n 1,, n r } be a reativey prime partition If n 1 is odd, then (n 1 1) is even and the right hand side in (49) is an integer, hence P Z n If n 1 is even, one of {n j : j = 2,, r} must be odd (since n 1, n 2,, n r are reativey prime) If n j is odd, then (by (42)) N ( 1 n n j ) 2Z + 1, which impies that N = 2N, and the right hand side in (49) is again an integer Hence, P is an integer vector Coroary 44 Let Π be a permutation matrix such that ˆP Π T P has entries in non decreasing order Then the matrix  ΠT AΠ is symmetric with respect to the anti diagona (i, n i + 1), i = 1,, n Proof We aready know that  = η 1(E ˆP T ˆP E T ) Because of the structure of P (Theorem 42), we concude that ˆP i = ˆP n i+1, i = 1,, n Hence, the entries â ij of  satisfy â ij = η 1 ( ˆP j ˆP i ) = η 1 ( ˆP n i+1 ˆP n j+1 ) = â n j+1,n i+1 Exampe 45 For instance, in the case of A = MSR(s) for s such as in Exampe 41, we ceary see the symmetry respect to the anti diagona which is based on the symmetry of the corresponding tupe (s Kac 1, s Kac 2,, s Kac 11 ) = (3, 5, 16, 9, 15, 0, 15, 9, 16, 5, 3) Acknowedgements We are greaty indebted to Mirko Primc for encouraging comments and discussions This work has been supported by Croatian Science Foundation under the project no 2634

17 SPECTRAL DECOMPOSITION OF A MATRIX OF SPECIALIZED ROOTS 71 References [1] R Bhatia, Matrix Anaysis, Springer, Graduate Texts in Mathematics 169, 1997 [2] N Bourbaki, Groupes et agébres de Lie, Chapitres 4,5,6, Hermann, 1968 [3] N Bourbaki, Groupes et agébres de Lie, Chapitres 7,8, Hermann, 1975 [4] J Gaier and D Xu, Computing exponentias of skew-symmetric matrices and ogarithms of orthogona matrices, Internationa Journa of Robotics and Automation 17 (2002), 1 11 [5] S Hegason, Differentia Geometry, Lie Groups, and Symmetric spaces, Academic Press, 1978 [6] J E Humphreys, Introduction to Lie agebras and Representation Theory, Springer- Verag, 1972 [7] V G Kac, Automorphisms of finite order of semi simpe Lie agebras, Funct Ana App 3 (1969), [8] V G Kac, Infinite dimensiona Lie agebras, Cambridge University Press, 1990 [9] V G Kac, D A Kazhdan, J Lepowsky and R L Wison, Reaisations of the basic representation of the Eucidean Lie agebras, Advance in Math 42 (1981), [10] V G Kac and D H Peterson, 112 constructions of the basic representation of the oop group of E 8, in: Symposium on anomaies, geometry, topoogy, Word Sci Pubishing, Singapore, 1985, [11] F ten Kroode and J van de Leur, Bosonic and fermionic reaization of the affine agebra ĝ n, Commun Math Phys 137 (1991), [12] J Lepowsky, Cacuus of twisted vertex operators, Proc Nat Acad Sci USA 82 (1985), [13] J Lepowsky and R L Wison, Construction of the affine Lie agebra A (1) 1, Comm Math Phys 62 (1978), [14] T Šikić, Z gradations of cassica affine Lie agebras and Kac parameters, Comm Agebra 32 (2004), O Kacovim parametrima i spektranoj dekompoziciji matrice specijaiziranih korijena Liejeve agebre s n Zatko Drmač i Tomisav Šikić Sažetak Ovaj rad prezentira zanimjiva spektrana svojstva odredene cjeobrojne antisimetrične matrice, koja nosi u sebi podatke o Z-gradacijama tipa s kasične afine Liejeve agebre sn Pokazano je da se skriveni Kacovi parametri mogu rekonstruirati iz ekspicitno izračunatih svojstvenih vektora pomoću Gram- Schmidtova postupka ortogonaizacije

18 72 Z DRMAČ AND T ŠIKIĆ Zatko Drmač Facuty of Sciences, Department of Mathematics University of Zagreb Zagreb Croatia E-mai: drmac@mathhr Tomisav Šikić Facuty of Eectrica Engineering and Computing University of Zagreb Zagreb Croatia E-mai: tomisavsikic@ferhr Received: