On the Minor Problem and Branching Coefficients
OOn the Minor Problemand Branching Coefficients
Jean-Bernard Zuber
Sorbonne Universit´e, UMR 7589, LPTHE, F-75005, Paris, France& CNRS, UMR 7589, LPTHE, F-75005, Paris, France
Abstract
The Minor problem, namely the study of the spectrum of a principal submatrix of a Hermi-tian matrix taken at random on its orbit under conjugation, is revisited, with emphasis on theuse of orbital integrals and on the connection with branching coefficients in the decompositionof an irreducible representation of U p n q , resp. SU p n q , into irreps of U p n ´ q , resp. SU p n ´ q . a r X i v : . [ m a t h . R T ] J un Introduction
What we call the Minor problem deals with the following question: given an n -by- n Hermitianmatrix of given spectrum, what can be said about the eigenvalues of one of its p n ´ q ˆ p n ´ q principal submatrices? This question has been thoroughly studied and answered by many authors[1–4]. As several other such questions, this problem of classical linear algebra has a counterpart inthe realm of representation theory [5–7], namely the determination of branching coefficients of anirreducible representation (irrep) of U p n q , resp. SU p n q , into irreps of U p n ´ q , resp. SU p n ´ q . Theaim of this note is to review these questions and to make explicit the link, by use of orbital integrals.It is thus in the same vein as recent works on the Horn [8–12] or Schur-Horn [13] problems.This paper is organized as follows. In sect. 2, I review the classical Minor problem and recallhow it may be rephrased in terms of U p n q orbital integrals. This suggests a modification, thatwill be turn out to be natural for the case of SU p n q . Sect. 3 is devoted to the issue of branchingcoefficients for the embeddings U p n ´ q Ă U p n q and SU p n ´ q Ă SU p n q . While the former is treatedby means of Gelfand–Tsetlin triangles and does not give rise to multiplicities, as well known sinceWeyl [14], the latter requires a new technique. This is where the modified integral introduced insect. 2 proves useful and is shown to provide an expression of branching coefficients, see Theorem2, which is the main result of this paper. The use of that formula as for the behaviour of branchingcoefficients under stretching , i.e., dilatation of the weights, is briefly discussed in the last subsection. Let us fix notations: If A is an n ˆ n Hermitian matrix with known eigenvalues α ě . . . ě α n ,what can be said about the eigenvalues β ě β ě . . . ě β n ´ of one of its principal p n ´ q ˆ p n ´ q minor submatrix (“minor” in short )?A first trivial observation is that if we are interested in the statistics of the β ’s as A is takenrandomly on its U p n q orbit O α , the choice of the minor among the n possible ones is immaterial,since a permutation of rows and columns of A gives another matrix of the orbit.A second, less trivial, observation is that the β ’s are constrained by the celebrated Cauchy–Rayleighinterlacing Theorem: α ě β ě α ě β ě . . . β n ´ ě α n . (1)For proofs, see for example [4, 15]If A is chosen at random on its orbit O α , and uniformly in the sense of the U p n q Haar measure,what is the probability distribution (PDF) of the β ’s ? This question has been answered byBaryshnikov [1], see also [3,4]. We first observe that the problem is invariant under a global shift ofall α ’s and all β ’s by a same constant: indeed a translation of A by a I n shifts by a all its eigenvaluesas well as all the eigenvalues of any of its principal minors.Let ∆ denote the Vandermonde determinant: ∆ n p α q “ ś ď i ă j ď n p α i ´ α j q and likewise for ∆ n ´ p β q . Theorem 1 (Baryshnikov [1]) . The PDF of the β ’s on its support (1) is given by P p β | α q “ p n ´ q ! ∆ n ´ p β q ∆ n p α q . (2)This result may also be recovered in terms of orbital integrals. Let H p n q α p X q “ ż U p n q dU e tr UαU : X (3) In the literature, the word “minor” refers either to the submatrix or to its determinant. We use here in theformer sense. X P H n , the space of n ˆ n Hermitian matrices, dU is the normalized Haar measure on U p n q ,and α stands here for the diagonal matrix diag p α i q . In terms of the eigenvalues x i of X , we writethis orbital integral as H p n q p α ; x q .The orbit O α carries a unique probabilistic measure, the orbital measure µ α p dA q , A P O α , whoseFourier transform is H p n q α p i X q E p e i tr AX q “ ż O α e i tr AX µ α p dA q “ ż U p n q dU e i tr UαU : X “ H p n q α p i X q . Let Π be the projector of H n into H n ´ that maps A P H n onto its upper p n ´ q ˆ p n ´ q minor submatrix B . According to the observation that the Fourier transform of the projection ofthe orbital measure is the restriction of the Fourier transform [4], the characteristic function of B is φ B p Y q “ φ A p X q , with X “ Π p X q “ ˆ Y
00 0 ˙ P H n , Y P H n ´ , from which the PDF of B isobtained by inverse Fourier transform P p B | A q “ p π q p n ´ q ż H n ´ dY e ´ i tr Y B ż U p n q dU e i tr UαU : X . After reduction to eigenvalues, P p β | α q “ p n ´ q ! p π q n ´ p ś n ´ p ! q ∆ n ´ p β q ż R n ´ dx ∆ n ´ p x q H p n q p α ; i p x, qq H p n ´ q p β ; i x q ˚ . (4)(In physicist’s parlance, this is the overlap of the two orbital integrals.) Here and below, for x P R n ´ , p x, q denotes the corresponding vector in R n .Making use of the explicit expressions known for H p n q p α ; x q [16, 17], we find P p β | α q “ C ∆ n ´ p β q ż R n ´ dx ∆ n ´ p x q det ` e i α i p x, q j ˘ i,j “ , ¨¨¨ ,n ∆ n p α q ∆ n pp x, qq det ` e ´ i β i x j ˘ i,j “ , ¨¨¨ ,n ´ ∆ n ´ p β q ∆ n ´ p x q (5) “ p π i q n ´ ∆ n ´ p β q ∆ n p α q ż R n ´ d n ´ xx x ¨ ¨ ¨ x n ´ det e i α i p x, q j det e ´ i β i x j (6)since the prefactor reads C “ p n ´ q ! ś n ´ p ! ś n ´ p ! i ´ n p n ´ q{ `p n ´ qp n ´ q{ p π q n ´ p ś n ´ p ! q “ p π i q n ´ and since ∆ n pp x, qq “ p ś n ´ i “ x i q ∆ n ´ p x q . Let’s write P p β | α q “ ∆ n ´ p β q ∆ n p α q K p α ; β q , hence K p α ; β q “ p n ´ q ! p π q n ´ p ś n ´ p ! q ∆ n p α q ∆ n ´ p β q ż R n ´ dx ∆ n ´ p x q H p n q p α ; i p x, qq H p n ´ q p β ; i x q ˚ “ p π i q n ´ ż R n ´ d n ´ xx x ¨ ¨ ¨ x n ´ det ` e i α i p x, q j ˘ ď i,j ď n det ` e ´ i β i x j ˘ ď i,j ď n ´ (7)in analogy with the introduction of the “volume functions” in the Horn and Schur problems [10,11, 13]. The function K p α ; β q is then, as in these similar cases, a linear combination of products ofDirichlet integrals: PP ş e i αt t “ i πε p α q , with ε the sign function. Thus K p α ; β q must be a piecewiseconstant function, supported by the product of intervals given by the interlacing theorem (1). The factor p n ´ q ! comes from the fact that we are restricting the β ’s to the dominant sector β ě β ě ¨ ¨ ¨ ě β n ´ .
2y making use of the integral form of the Binet–Cauchy formula, (see [4]), namely ż R k det p f i p t j qq ď i,j ď k det p g i p t j qq ď i,j ď k “ k ! det ˆż R f i p t q g j p t q ˙ ď i,j ď k , with here k “ n ´ f i p t q “ t p e i α i t ´ e i α n t q , g i p t q “ e ´ i β i t , we find K p α ; β q “ p n ´ q ! p π i q n ´ det ż R dtt p e i p α i ´ β j q t ´ e i p α n ´ β j q t q (8) “ p n ´ q !2 n ´ det ` ε p α i ´ β j q ´ ε p α n ´ β j q ˘ ď i,j ď n ´ (9) “ p n ´ q !2 n ´ det ¨˚˚˚˝ ε p α ´ β q ¨ ¨ ¨ ε p α ´ β n ´ q ε p α ´ β q ¨ ¨ ¨ ε p α ´ β n ´ q ε p α n ´ β q ¨ ¨ ¨ ε p α n ´ β n ´ q ˛‹‹‹‚ . (10)Equ. (9) just reproduces a result by Olshanksi [3], since the difference ` ε p α i ´ β j q ´ ε p α n ´ β j q ˘ appearing there is nothing else than twice the characteristic function of the interval r α n , α i s , denoted M p β j ; α n , α i q in [4]. Finally, it may be shown that the determinant in (10) equals 2 n ´ times thecharacteristic function of (1), so that the piecewise constant function K is just p n ´ q ! on itssupport, in agreement with (2), see [3, 4]. Remark . The previous considerations extend to projections of matrix A onto a smaller minor k ˆ k submatrix, see [2–4]. In this section, we introduce a modification of the integral (7) that will be suited in our later studyof SU p n q branching rules. We first change variables in K , introducing the spacings γ i “ β i ´ β n ´ , i “ , ¨ ¨ ¨ , n ´ K p α ; β q “ p π i q n ´ ż R n ´ d n ´ xx x ¨ ¨ ¨ x n ´ e ´ i β n ´ ř n ´ j “ x j det ` e i α i p x, q j ˘ ď i,j ď n det ` e ´ i γ i x j ˘ ď i,j ď n ´ , where by convention γ n ´ “
0. We then integrate over β n ´ (while introducing a 1 {p n ´ q ! prefactorfor later convenience), and define s K p α ; γ q : “ p n ´ q ! ż dβ n ´ K p α ; γ ` β n ´ q (11) “ p π qp n ´ q ! p π i q n ´ ż R n ´ d n ´ xx x ¨ ¨ ¨ x n ´ δ ´ n ´ ÿ x j ¯ det ` e i α i p x, q j ˘ ď i,j ď n det ` e ´ i γ i x j ˘ ď i,j ď n ´ . Remark . Although it depends only on the spacings γ , the function s K is not directly related tothe PDF of the spacings in the original Minor problem. Its introduction is rather motivated by itsconnection with the su p n q Lie algebra, see below sect. 3.2.Thus integrating K over β n ´ amounts to considering a modified integral, where in (7) weintegrate on the p n ´ q -dimensional hyperplane ř n ´ i “ x i “
0. Hence an alternative definition of s K is s K p α ; γ q “ ∆ n p α q ∆ n ´ p γ qp π q n ´ p ś n ´ p ! q ż R n ´ dx δ ´ n ´ ÿ x j ¯ ∆ n ´ p x q H p n q p α ; i p x, qq H p n ´ q p γ ; i x q ˚ , (12)3n expression that we use later in sect. 3.2. A more explicit expression is s K p α ; γ q “ π p π i q n ´ p n ´ q ! ż ř n ´ x i “ dxx x ¨ ¨ ¨ x n ´ det p e i αix ,e i αix , ¨¨¨ ,e i αixn ´ , q ď i ď n ´ det ` e ´ i γ i x j ˘ ď i,j ď n ´ (13)and we note that, because of the constraint ř n ´ i “ x i “
0, this expression is invariant by a globalshift of all α i . We may use that invariance to choose α n “
0, a choice that will be natural inthe application to SU p n q representations. We conclude that s K p α, γ q is a function of two sets ofvariables, a n -plet α with α n “
0, and a p n ´ q -plet γ with γ n ´ “
0. As is clear from (11), s K may be extended to a function of the unordered α ’s and γ ’s, odd under the action of the symmetricgroup S n , i.e., the SU p n q Weyl group, acting on α by w p α q i “ α w p i q ´ α w p n q , i “ , ¨ ¨ ¨ , n , w P S n ,and likewise odd under the action of S n ´ on γ i , i “ , ¨ ¨ ¨ , n ´ ş e i at t r “ i π p i a q r ´ p r ´ q ! ε p a q , one finds that s K p α ; γ q , a combination of convoluted box splines, is a piece-wiselinear function of differentiability class C . Its support is the polytope defined by the inequalities(recall that by convention α n “ ď i ď n ´ p α i ` ´ γ i q ď min ď i ď n ´ p α i ´ γ i q (14)that guarantee that there exist β n ´ satisfying the simultaneous inequalities (1), i.e., α i ` ď β i “ γ i ` β n ´ ď α i , for all i “ , ¨ ¨ ¨ , n ´ . (15)The maximal value (in the dominant sector) of s K , for fixed α , is readily derived from (11), wherewe are integrating the function K {p n ´ q ! equal to 1 on its support, over β n ´ , subject to the n ´ γ s K p α ; γ q “ min ď i ď n ´ p α i ´ α i ` q . (16)Let (cid:15) w denote the signature of permutation w P S n . For n “ s K reads s K p α ; γ q “ ´ | α ´ γ | ´ | α ´ α ´ γ | ´ | α ´ γ | ¯ ´ p γ ÞÑ ´ γ q (17) “ ÿ w P S (cid:15) w | w p α q ´ γ | which is an odd continuous function of γ , vanishing for γ R p´ α , α q , constant and equal to itsextremum value ˘ min p α ´ α , α q for | γ | P r min p α ´ α , α q , max p α ´ α , α qq , and linear inbetween, see Fig. 1.For n “
4, let w p ¯ α q i “ α w p i q ´ α w p n q , ψ p α ; γ q : “ ε p α ´ γ q ´ | α ´ γ | ´ | α ´ γ | ´ | α ´ α ´ γ ` γ | ` | α ´ α ´ γ ` γ | ¯ (18)then s K p α ; γ q “ ÿ w P S (cid:15) w ψ p w p α q ; γ q (19) s K p α ; γ q has a support in the dominant sector defined by the inequalities p α ´ α q ď γ ď α , ď γ ď α , ď γ ´ γ ď α ´ α , (20)and a maximal value equal to min p α ´ α , α ´ α , α q . Its graph has an Aztec pyramid shape,see Fig. 2. 4 γ K Figure 1: The s K function for n “ α “ t , , u and γ ě γ “ s K function for n “ α “ t , , , u , in the γ ě γ ě γ “ In this section, we consider the restriction of the group U p n q , resp. SU p n q , to its subgroup U p n ´ q ,resp. SU p n ´ q , and the ensuing decomposition of their representations. For definiteness, therestriction of SU p n q to SU p n ´ q we have in mind results from projecting out the simple root ααα n ´ in the dual of the Lie algebra su p n q , and likewise for U p n q . Just like in the cases of the Horn or of the Schur problem, the Minor problem is the classicalcounterpart of a “quantum” problem in representation theory. Given a highest weight (h.w.)irreducible representation (irrep) V p n q α of U p n q , which irreps V p n ´ q β of U p n ´ q occur and withwhich multiplicities, in the restriction of U p n q to U p n ´ q ? That problem too is well known, isimportant in physical applications (see for example [18, 19]), and may be solved by a variety ofmethods. Here we first recall how to make use of Gelfand–Tsetlin triangles, i.e., triangular patterns x p n q “ α ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ x p n q n “ α n x p n ´ q ¨ ¨ ¨ ¨ ¨ ¨ x p n ´ q n ´ . . . . .. x p q subject to the inequalities x p j ` q i ě x p j q i ě x p j ` q i ` , ď i, j ď n ´ . (21)5n the present context, the α ’s denote the lengths of the rows of the Young diagram associatedwith the irrep V p n q α . The number of solutions of the inequalities (21) gives the dimension of theirrep dim p V p n q α q “ t x p j q i | solutions of p qu . The values β i “ x p n ´ q i , 1 ď i ď n ´
1, appearing in the second row of the triangle give the lengthsof rows of the Young diagrams of the possible representations V p n ´ q β of U p n ´ q . Given thosenumbers, the number of solutions x p j q i , 1 ď i, j ď n ´ p n ´ q . Thus we have the sum-ruledim p V p n q α q “ t x p j q i | solutions of p q , x p n q “ α u (22) “ ÿ β t x p j q i | solutions of p q , x p n q “ α, x p n ´ q “ β u “ ÿ β dim p V p n ´ q β q , which is consistent with the multiplicity 1 of each V p n ´ q β appearing in the decomposition, a classicalresult in representation theory [14, 20, 21], see also chapter 8 in [22]. Thus one sees that the β ’ssatisfy the inequalities (1) and one may say that the branching coefficient, equal to 0 or 1, is givenbr α p β q “ K p α ; β q (23)with the convention that the discontinuous function K is assigned the value 1 throughout its support,including its boundaries.Going from U p n q to SU p n q , we have to restrict to Young diagrams with less than n rows, orequivalently, to reduce Young diagrams with n rows by deleting all columns of height n . Startingfrom an irrep of SU p n q , we apply to it the procedure above, and then remove the columns of height n ´ p q . Take for α the adjoint representation, i.e., α “ t , , u . The possible β satisfying (1) are written in red in what follows2 ě ě ě ě . e ., β “ t , u ” t , u (24)2 ě ě ě ě . e ., β “ t , u ” t , u (25)2 ě ě ě ě . e ., β “ t , u (26)2 ě ě ě ě . e ., β “ t , u (27)where two β are regarded as equivalent if their Young diagrams differ by a number of columns ofheight n ´ “
2. Hence in SU p q Ă SU p q , we writebr α p β q “ , , β “ t , u , t , u , t , u (28)and we check the sum-rule on dimensions: 8 “ ` ˆ `
3. Note that removing columns ofheight n ´ β amounts to focusing on spacings between the β ’s, which points to the relevance of our function s K .To summarize, in the “quantum”, U p n q -representation theoretic, problem, the β ’s are the in-teger points interlacing the α ’s and come with multiplicity 1, while in the SU p n q case, non trivialmultiplicities may occur and the interlacing property no longer applies. In the next subsection, weshow that the latter multiplicities are given by the function s K of sect. 2.2.6 .2 A s K ´ br relation The multiplicities occurring in the SU p n ´ q Ă SU p n q problem may be expressed in terms ofcharacters by the integral br α p γ q “ ż Dt χ p n q α p e i p t, q q ` χ p n ´ q γ p e i t q ˘ ˚ (29)which computes the projection of the SU p n q character χ p n q α restricted to the Cartan torus of SU p n ´ q onto the SU p n ´ q character χ p n ´ q γ . There, Dt stands for the Haar measure on the Cartan torus T n ´ of SU p n ´ q Dt “ | p ∆ n ´ p e i t q| p π q n ´ p n ´ q ! dt where we use the notations p ∆ n ´ p e i t q : “ ź ααα ą ` e i x ααα,t y{ ´ e ´ i x ααα,t y{ ˘ , and ∆ n ´ p t q : “ ź ααα ą x ααα, t y ,ααα the positive roots of su p n ´ q , and dt is the Lebesgue measure on T n ´ . Theorem 2.
The branching coefficient, that gives the multiplicity of the irrep of SU p n ´ q of h.w. γ in the decomposition of the irrep of SU p n q of h.w. α , is br α p γ q “ s K p α ` ρ n ; γ ` ρ n ´ q (30) with ρ n the Weyl vector of the algebra su p n q , and ρ n ´ that of su p n ´ q .Proof . We recall Kirillov’s relation between a SU p n q character and the orbital integral: χ p n q α p e i t q “ dim V α ∆ n p i t q p ∆ n p e i t q H p n q p α ` ρ n ; i t q (31)with dim V α “ ∆ n p α ` ρ n q ∆ n p ρ q . Plugging in (12) the expression (31) and the analogous one for su p n ´ q leads to s K p α ` ρ n ; γ ` ρ n ´ q “ ∆ n p α ` ρ n q ∆ n ´ p γ ` ρ n ´ qp π q n ´ p ś n ´ p ! q ż R n ´ dt ∆ n ´ p t q H p n q p α ` ρ n ; i p t, qq H p n ´ q p γ ` ρ n ´ ; i t q ˚ “ ś n ´ p “ p ! ś n ´ p “ p ! p π q n ´ p ś n ´ p ! q ż R n ´ dt ∆ n ´ p t q p ∆ n p e i p t, q q p ∆ n ´ p e i t q ˚ ∆ n p i p t, qq ∆ n ´ p i t q ˚ χ p n q α p e i p t, q q χ p n ´ q γ p e i t q ˚ “ i ´p n ´ q p π q n ´ p n ´ q ! ż R n ´ dt | p ∆ n ´ p e i t q| χ p n q α p e i p t, q q χ p n ´ q γ p e i t q ˚ ∆ n ´ p t q ∆ n pp t, qq p ∆ n p e i p t, q q p ∆ n ´ p e i t q“ i ´p n ´ q ż T n ´ Dt ÿ δ P πQ _ χ p n q α p e i p t ` δ, q q χ p n ´ q γ p e i p t ` δ q q ˚ ∆ n ´ p t ` δ q ∆ n pp t ` δ, qq p ∆ n p e i p t ` δ, q q p ∆ n ´ p e i p t ` δ q q where the integration is now carried out on the Cartan torus of SU p n ´ q , T n ´ “ R n ´ {p πQ _ q , Q _ the p n ´ q -dimensional coroot lattice of SU p n ´ q , which is, in the present simply laced case,isomorphic to the root lattice. Only the ratio ∆ n ´ p t ` δ q ∆ n pp t ` δ, qq depends on δ and the summation can becarried out with the result thati ´p n ´ q p ∆ n p e i p t, q q p ∆ n ´ p e i t q ÿ δ P πQ _ ∆ n ´ p t ` δ q ∆ n pp t ` δ, qq “ . (32)7ndeed, if we write p t, q in the su p n q root basis: p t, q “ ř n ´ j “ a j ααα j (with no component on ααα n ´ q , p ∆ n p e i p t, q q p ∆ n ´ p e i t q “ ´p q n ´ sin a ˜ n ´ ź i “ sin a i ` ´ a i ¸ sin a n ´ , (33)(on which it is clear that it is invariant under a i ÞÑ a i ` p i p π q , p i P Z ), while∆ n pp t, qq ∆ n ´ p t q “ ´ a ˜ n ´ ź i “ p a i ` ´ a i q ¸ a n ´ (34)and the identity (32) follows from a repeated use of ÿ p “´8 p a ` πp qp b ´ a ´ πp q “ b sin b sin a sin b ´ a in the telescopic product (34). l Remark . The proof above follows closely similar proofs in [10, 13] that relate the classicalHorn or Horn–Schur problems to the computation of Littlewood–Richardson or Kostka coefficients.However, in contrast with those cases, here the r.h.s is a single term, rather than a linear combi-nation involving a convolution.Together with the results of the end of sect. 2.2, Theorem 2 has immediate consequences:
Corollary 1.
The number of irreps of SU p n ´ q appearing in the decomposition of the irrep of SU p n q of h.w. α (with α n “ ) is equal to the number of integer points in the polytope defined bythe inequalities (14), where α is changed into α ` ρ n , namely t γ P Z n ´ ` | γ ě ¨ ¨ ¨ ě γ n ´ ě , max ď i ď n ´ p α i ` ` n ´ i ´ ´ γ i q ď min ď i ď n ´ p α i ` n ´ i ´ γ i qu . Here as before, the α i are the Young coordinates of the h.w. α , i.e., , the lengths of the rows of itsYoung diagram.On the other hand, eq. (30) together with (16) gives the maximal value of a branching coefficientof a given α max γ br α p γ q “ min i pp α ` ρ n q i ´ p α ` ρ n q i ` q “ min i p α i ´ α i ` q ` α i ´ α i ` is just the i -th Dynkin component of the weight α . Corollary 2.
The largest multiplicity (branching coefficient) that occurs in the branching of anirrep of SU p n q of h.w. α into irreps of SU p n ´ q is 1 plus the smallest Dynkin component of α . Examples . Take n “ α “ t , , u , i.e., p , q in Dynkincomponents, α ` ρ “ t , , u , γ P tt , u , t , u , t , uu , γ ` ρ P tt , u , t , u , t , uu , one findswith the formula (17): s K p α ` ρ ; γ ` ρ q “ , , n “
4, take α “ t , , , u , i.e., p , , q in Dynkin components, one finds the following decom-position into 18 SU p q weights t , , , u“p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q (36) ‘p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q ‘ p , q in terms of Dynkin components, and with the multiplicity appended as a subscript. Recall that the Dynkin components of a weight are its components in the fundamental weight basis. Hereafter,they are denoted by round brackets. .3 Stretching The relation (30) is also well suited for the study of the behaviour of branching coefficients under“stretching”. From (35) we learn that the growth is at most linearbr sα p sγ q ď s min p α i ´ α i ` q ` . For example, for n “
3, with Dynkin components, br p s,s q p s q “ s ` p s,s q p s q “ s K pt s ` , s ` , u ; t s ` , uq “ p s ` q s K pt , , u ; t , uq “ s ` p s,s q p s ´ q or br p s,s q p s q , we are not probing the function on its plateau and its behaviouris not always linear in s : s K pt s ` , s ` , u ; t γ , uq “ γ if 0 ď γ ď s ` p s ` q ´ γ if s ` ď γ ď p s ` q whence br p s,s q p s ´ q “ s K pt s ` , s ` , u ; t s, uq “ s and br p s,s q p s q “ s K pt s ` , s ` , u ; t s ` , uq “ γ γ γ γ γ γ Figure 3: Weights in the γ -plane (in Dynkin components) appearing in the decomposition of theweight α “ s t , , , u ” s p , , q of SU p q , for s “ , ,
3. Markers of different colourscode for multiplicities from 1 to 4.Figure 4: The s K function for n “ α ` ρ “ t , , , u “ t , , , u`t , , , u and γ , γ areDynkin components. The cross-sections at altitude 1 , , , p q Ă SU p q , the points of increasing multiplicity form a matri-ochka pattern, see Fig. 3, in a way already encountered in the Littlewood–Richardson coefficientsof SU p q , [23]. This pattern just reproduces the cross-sections of increasing altitude of the Aztecpyramid of Fig. 4. Acknowledgements
All my gratitude to Robert Coquereaux for his constant interest, encouragement and assistance. Ialso want to thank Jacques Faraut and Colin McSwiggen for their critical reading and comments.
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