The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices

Horn's problem, i.e., the study of the eigenvalues of the sum C=A+B of two matrices, given the spectrum of A and of B, is re-examined, comparing the case of real symmetric, complex Hermitian and self-dual quaternionic 3 x 3 matrices. In particular, what can be said on the probability distribution function (PDF) of the eigenvalues of C if A and B are independently and uniformly distributed on their orbit under the action of, respectively, the orthogonal, unitary and symplectic group ? While the two latter cases (Hermitian and quaternionic) may be studied by use of explicit formulae for the relevant orbital integrals, the case of real symmetric matrices is much harder. It is also quite intriguing, since numerical experiments reveal the occurrence of singularities where the PDF of the eigenvalues diverges. Here we show that the computation of the PDF of the symmetric functions of the eigenvalues for traceless 3 x 3 matrices may be carried out in terms of algebraic functions --roots of quartic polynomials-- and their integrals. The computation is carried out in detail in a particular case, and reproduces the expected singular patterns. The divergences are of logarithmic or inverse power type. We also relate this PDF to the (rescaled) structure constants of zonal polynomials and introduce a zonal analogue of the Weyl SU(n) characters. This version (V2) : minor corrections, one figure and one reference added.


Introduction
Recall what Horn's problem is: given two nˆn matrices A and B of given spectrum of eigenvalues, what can be said about the spectrum of their sum C " A`B? The problem has been addressed by many authors, see [1] for a review and references. For Hermitian matrices, the support of the spectrum of C has been completely determined after some crucial work by Klyashko [2] and by Knutson and Tao [3,4,5]. In a recent paper [6], a more specific question was considered: given α " tα 1 ě α 2 ě¨¨¨α n u and β " tβ 1 ě β 2 ě¨¨¨β n u, take the Hermitian matrices A and B uniformly and independently distributed on the orbit of diag pαq and diag pβq under the action of the SUpnq group. The probability distribution function (PDF) of the eigenvalues γ of C " A`B may then be computed, see also [7,8].
The same question may, however, be raised if instead of Hermitian matrices, one considers other classes of matrices and the action of an appropriate group. The case of skew-symmetric real matrices was considered in [6], but more interesting is that of real symmetric matrices and the action of the orthogonal group SOpnq.
Let us start with a numerical experiment with specific 3ˆ3 matrices. Since tr C " tr A`tr B, only two eigenvalues of C, say γ 1 and γ 2 are linearly independent. Then compare the distribution of points in the pγ 1 , γ 2 q-plane for the three cases of (a) orbits of real symmetric matrices A and B equivalent to J z :" diag p1, 0,´1q under conjugation by the orthogonal group SOp3q; (b) orbits of such matrices, regarded as Hermitian, under conjugation by the unitary group SUp3q; (c) orbits of such matrices, regarded as quaternionic self-dual, (i.e., A b I 2 , B b I 2 as 6ˆ6 matrices), under conjugation by the unitary symplectic group USpp3q. Following the nomenclature introduced by Dyson, we label these three cases by the index β " 1,2 or 4, respectively 1 .
Some features appear clearly on the plots and histograms of Fig. 1: (i) the PDF vanishes faster on the boundaries of the Horn domain as β increases; (ii) the non-analyticities are stronger and stronger as β decreases; (iii) these singularities seem to appear at the same place in the pγ 1 , γ 2 q plane (for α and β fixed). In the Hermitian case (and in the quaternionic case as well, see below), it is known that the PDF is a piece-wise polynomial function. The plots of Fig. 1 (a) show that this cannot be true for real symmetric matrices. It should be emphasized that these general features do not depend on the explicit case we have chosen. Similar singular patterns have been observed in numerical experiments with other matrices A and B, see Fig. 7 in [6].
The aim of this paper is to compare the three cases, to reproduce analytically the previous empirical observations and in particular to analyse the location and nature of the singularities that occur in the symmetric case. After a brief review of known results on the relevant orbital integrals (sect. 1), we treat rapidly the easy case of quaternionic self-dual matrices (sect. 2), before turning to the more challenging case of real symmetric matrices in sect. 3 and 4. In sect. 3, it is shown that for 3ˆ3 traceless matrices, the introduction of the two symmetric functions p and q of the three eigenvalues (of vanishing sum) simplifies matters: one may express the PDF ρpp, qq in terms of roots of some polynomial equations and integral thereof, see (33) below. The actual computation is carried out again for our pet example of α " β " p1, 0,´1q in sect. 4. In particular we reproduce and analyze in sect. 4.4 and 4.5 the singularities that are apparent in Fig. 1 (a). Finally in sect. 5, we show that this function ρpp, qq is related to the distribution of the (rescaled) structure constants of zonal polynomials, thus elaborating on a claim of [9]. 1 The orbital integrals for β " 1, 2, 4 Consider the orbital integrals 1. β " 1, Ω P G 1 :" SOpnq, X, A real symmetric matrices; 2. β " 2, Ω P G 2 :" Upnq, X, A complex Hermitian matrices; 3. β " 4, Ω P G 4 :" USppnq, X, A real quaternionic self-dual matrices.
These closed formulae allow an explicit computation of the PDF, see [6] and below, sect. 2. In contrast, for β " 1, i.e., for real symmetric matrices under the action of the orthogonal group SOpnq, the best that can be achieved is an expansion in zonal polynomials [12,13,9] I 1 px, αq " where the second sum runs over partitions κ of m with no more than n parts, and cpκq is a constant which depends on the normalization of the Zpκq's, see sect. 5. If the zonal polynomials in the above formula are written with the so-called James normalization, one can uses the values of ZpκqpIq and of the coefficient cpκq tabulated by James up to m " 4, for all n (the dimension of I) in the Appendix of [12], p.157; one can also use commands contained in the Mathematica package [14]. The infinite sum in the previous expansion, however, makes the computation of the PDF intractable (as far as we can see), and we will have to follow another route, see below section 3.
In section 5, however, we return to this formulation in terms of zonal polynomials and show a connection between the PDF and the distribution of (rescaled) "zonal multiplicities" i.e., appropriate structure constants of zonal polynomials.
2 Quaternionic case for n " 3 We start with a warming up exercise: compute the PDF of the eigenvalues γ for a sum of two self-dual quaternionic matrices that are independently and uniformly distributed on their orbit under the action of the unitary symplectic group. The computation of the PDF of the γ's follows the same lines as that in the Hermitian case. We will therefore be a bit sketchy in its derivation, referring the reader to [6] for details of the computation.
Up to an overall factor, this PDF is given by the integral of three orbital integrals of the type where N " np2n´1q is the number of independent matrix elements of a self-dual quaternion matrix and κ n " p2πq pN´nq{2 n!p2n´1q!p 1 2 npn´1qq! stems from the change from those N variables to the n eigenvalues. Thus e i u j A j pP,P 1 ,Iq f n pτ pi α, P qqf n pτ pi β, P 1 qqf n pτ p´i γ, Iqq with u j :" x j´xj`1 , ∆ 1 puq :" ś 1ďiăjďn pu i`ui`1`¨¨¨uj´1 q and A j pP, P 1 , P 2 q " j ÿ k"1 pα P pkq`βP 1 pkq´γP 2 pkq q .  Figure 2: Comparing the histogram of 10 4 points with the PDF of (10) for (left) α " β " p1,´1q and (right) α " p1,´1q, β " p2,´2q.
With the little extra complication caused by the f n factors of (3), the integration may be carried out as in [6], namely by partial fraction decomposition of the integrand and repeated use of the Dirichlet formula for the Cauchy principal value with the sign function. The result, though cumbersome, is completely explicit. For n " 2, we leave as an exercise for the reader to check that for α 2 "´α 1 , β 2 "´β 1 , we have ppγ|α, βq " 6 ∆pγq ∆ 3 pαq∆ 3 pβq δpγ 1`γ2 qJ 2 with showing that the spectrum of γ 1 is supported by two segments I :" r|α 1´β1 |, α 1`β1 s and´I, or only by the former if one imposes γ 2 ď γ 1 . Compare with the analogous formulae obtained for Hermitian and real symmetric matrices in [6]. In Fig. 2, we show the agreement with the histogram made of 10 4 for two pairs α " β " p1,´1q and α " p1,´1q, β " p2,´2q. For n " 3, where J 3 is a piecewise polynomial of degree 13 in γ 1 and γ 2 ; indeed (F pu 1 , u 2 q stands for the product f 3 pτ pi α, P qqf 3 pτ pi β, P 1 qqf 3 pτ p´i γ, Iqq in (8), see also (4)), whence, by homogeneity, a behavior as rγs 15´2"13 . J 3 is of differentiability class C 2 : indeed as u 1 , say, goes to infinity, the integrand behaves as 1 u 4 1 e i A 1 u 1 (up to subdominant terms), so that the integral is twice continuously differentiable with respect to A 1 (or γ), hence of class C 2 . Non-analyticities are expected (and do occur) along the boundaries of the Horn polygon and across the same singular lines (or half-lines) as in the Hermitian case, namely the same with α Ø β . Figure 3: The four sectors in the Horn polygon of the quaternionic case for α " β " t1, 0,´1u.
For our favorite example of A " B " J z " diag p1, 0,´1q, the function has four sectors of piecewise polynomiality, labelled by i " 1,¨¨¨, 4 according to Fig. 3. In each of these four sectors the function takes the form where and each of thep i pγ 1 , γ 2 q " 40γ 10 1`¨¨¨i s too cumbersome to be given here 2 . Note that the form (15) guarantees that J 3 vanishes "cubically" on each boundary of the polygon. This explains one of the features observed on the plots of Fig. 1 (c). As a side remark, we note thatp 4 pγ 1 , γ 2 q is symmetric under γ 1 Ø γ 2 , andp 3 pγ 1 , γ 2 q is symmetric under γ 2 Ø γ 3 " γ 1´γ2 . One may also compute the transition functions between adjacent domains: where in each case, the ellipsis stands for a polynomial of degree 10 that we refrain from displaying. Just like on the boundary, the transition functions vanish cubically along the non-analyticity lines. This is in agreement with the differentiability argument above.
The resulting PDF is plotted in Fig. 4 (a) and compared to the "experimental" histogram of Fig. 4 (b).
A good check of our calculation is that the integral of the PDF over the whole domain of γ's, made of 3! copies of the Horn polygon r H α,β , (see below in (19)), is indeed 1, thanks to ş r H α,β dγ 1 dγ 2 ∆pγqJ 3 " 1 810 .
3 Computing the PDF for a sum for two real symmetric matrices As explained above, in contrast with the unitary or the symplectic groups, there is no closed expression for the orbital integral on the orthogonal group and the formula ppγ|α, βq " const. |∆pγq| remains intractable. We thus have to resort to a different approach. We first trade the PDF of eigenvalues γ i of the sum C " A`B for the PDF ρpp, qq of symmetric functions of the γ's, see (20) below. The relation between the two is given below in (24).

The support of the symmetric functions P and Q
For 3ˆ3 real symmetric traceless matrices A and B, the characteristic polynomial of their sum C reads 3 detpzI´Cq " detpzI´A´Bq " z 3`P z`Q .
It has been suggested [6] that the Horn problem for real symmetric matrices not only shares the same support as in the Hermitian case (for given α and β), as proved by Fulton [1], but has also singularities at the same locations, although the latter look much stronger. Thus we expect 1000 q Figure 5: The Horn-tensor polygon r H αβ in the pγ 1 , γ 2 q plane, and the curlinear polygon in the pp, qq plane, drawn here for α " t11,´1,´10u, β " t7, 4,´11u. The dashed (blue, red and green) lines are the expected loci of singularities of the PDF. (A histogram of eigenvalues γ for an equivalent configuration of pα, βq has appeared in [6], Fig. 7.) singularities to occur for 3ˆ3 matrices along the same lines (or half-lines) as in (13) 4 . These lines are illustrated in Fig. 5 for α " t11,´1,´10u, β " t7, 4,´11u in the pγ 1 , γ 2 q and in the pp, qq planes.

The statistics of the symmetric functions P and Q
For 3ˆ3 real traceless symmetric matrices A and B, of respective eigenvalues α and β, the characteristic polynomial of their sum C reads detpz I 3´C q " detpz I 3´d iag pαq´R diag pβqR T q " z 3`P pRqz`QpRq .
Note the peculiar feature of these polynomials in c: their degree 0 and 2 terms are (degree 1) polynomials in the variables x 2 and y 2 , while their degree 1 term in c is of the form xy sin φ sin ψ, up to a (α-and β-dependent) factor.
Note also that these expressions are π-periodic in φ and ψ, making it possible to restrict these angles to the interval p0, πq where their sine is non negative. This will be implicit in the following. Thus The PDF for the independent variables γ 1 , γ 2 then follows simply: 3.3 Reducing δpQ q qδpP p q to δpRq In this subsection, we show that ş dc δpQ q qδpP p q may be reduced, up a factor, to a single δpRq, where R is the resultant of the two polynomials P p pcq and Q q pcq. We shall make repeated use of two classical identities [15]: -for f ptq a function with a finite number of "simple" zeros t i (i.e., such that f 1 pt i q ‰ 0), -for f and g two functions with no common zero, to which we may then apply the previous identity. Then starting from the product δpQ q qδpP p q, we assume that the discriminant ∆ Q of Q q is positive, in such a way that the roots c 1,2 are real and distinct, and we may write and ż dc δpQ q qδpP p q " 1 a ∆ Q pδpP p pc 1 qq`δpP p pc 2 qqq (where it is understood that the delta's act on functions of the remaining variables φ and ψ or x and y).
We want to compare this expression with δpRq where, as said above, R is the resultant of the two polynomials P p pcq and Q q pcq. If a and a 1 are the coefficients of terms of degree 2 of the polynomials Q q and P p , respectively, and c 1 1 , c 1 2 the roots of the latter, thus P p pcq " a 1 pc´c 1 1 qpc´c 1 2 q Q q pcq " apc´c 1 qpc´c 2 q , the resultant R defined as a 2 a 12 ś i,j"1,2 pc i´c 1 j q may also be written as R " a 2 P p pc 1 qP p pc 2 q .
(For the polynomials of (22), this is a quite cumbersome polynomial of degree 5 in u " x 2 and in z " y 2 , with 4089 α-and β-dependent terms 2 .) According to (26), one writes But P p pc 1,2 q has the general form with A and B functions of x and y. Thus whenever P p pc 1 q vanishes for some px, yq, i.e., A " B a ∆ Q , we have P p pc 2 q "´2B a ∆ Q for those values. And vice versa, if P p pc 2 q vanishes, then P p pc 1 q " 2B a ∆ Q . One may thus rewrite (27) as a non trivial and useful identity. For the polynomials P p and Q q of (22), it is easy to compute that 2a 2 B " pxy sin φ sin ψqB, with B " 2pα 1´α2 qpβ 1´β2 qˆ (29) rp2α 1`α2 qpα 1`2 α 2 qppβ 1´β2 qz`pβ 1`2 β 2 qq`p2β 1`β2 qpβ 1`2 β 2 qppα 1´α2 qu`pα 1`2 α 2 qqs while the prefactor |x| |y| sin φ sin ψ (recalling that x " cos φ, y " cos ψ) enables us to change variables to u " x 2 , z " y 2 , with the result that The final transformation of this expression follows from (25). If u i pzq denote those roots of the polynomial Rpu, zq that belong to the interval r0, 1s, we may write This integral will be studied more explicitly in a particular example in the next section. Three remarks are in order: -we have assumed from the start that the discriminant ∆ Q is positive, and this led us to (30). Conversely, the vanishing of R for real values of the variables u and z encompassed in (30) implies that P p pcq and Q q pcq have a common root, and this may only be possible if that root is a root of aP p´a 1 Q q which is a degree 1 polynomial in c with real coefficients. Thus the common roots of P p pcq and Q q pcq are necessarily real, justifying our assumption that ∆ Q ě0; -the roots c 1 and c 2 have to lie in r´1, 1s for the consistency of the derivation; -the two functions P p pc 1 q and P p pc 2 q have to have no common zero. Otherwise if that happened at some values px, yq, both A and B would vanish. The two latter points will be verified in the particular case that we discuss now.  Figure 6: Left, the curlinear polygon, bounded by the lines 2p`8˘q " 0 and two arcs of the cubic pp{3q 3`p q{2q 2 " 0; (right) the distribution of 8000 points in the pp, qq plane for A " B " diag p1, 0,´1q. The (red and black) dashed lines along which singularities are expected are q " 0 and p "´1˘q 4 The particular case A " B " J z

Symmetries and reduction to algebraic equations
From now on, we restrict ourselves to the particular case of α " β " t1, 0,´1u. The polynomials P p and Q q then reduce to where we recall that c " cos θ, x " cos φ, y " cos ψ, and we could substitute sin φ " ? 1´x 2 , sin ψ " a 1´y 2 since φ and ψ are restricted to p0, πq. Note that these expressions are invariant under the exchange of φ and ψ, or of x and y.
Another symmetry of the problem will be quite useful. Because´J z is conjugate to J z by the action of R y pπ{2q, it is clear that the distribution of the γ's will be invariant under change of sign, and the distribution of the pp, qq variables will be invariant under change of sign of q. This is apparent in Fig. 6 where we see that both the support of ρ, here a curvilinear quadrangle, and the distribution of points (here a simulation with 8000 points), are symmetric under q Ñ´q. Our formalism, however, does not have that manifest symmetry, and we will find it useful in the following to choose q negative, as we see shortly. The PDF for all values of q will then be reconstructed by symmetry.
The discriminant of Q q is ∆ Q " 16p1´x 2 qp1´y 2 q`8qp1´x 2 y 2 q. As explained previously, its positivity follows by consistency from the vanishing of the resultant R. For´2 ď q ď 0, one may check that, if ∆ Q ě 0, the real roots obey |c 1,2 | ď 1 for all x, y P r´1, 1s. We shall hereafter assume that´2 ď q ď 0.
Following the discussion of sect. 3.3, we then determine the resultant of the two polynomials P p pcq and Q q pcq, a degree 4 polynomial in u and z Of course, this polynomial R is also symmetric under the swapping of u and z, since P and Q were under x Ø y. The factorB appearing in (30) isB " 4p2`x 2`y2 q. Then, according to (31), the PDF reads

Roots u i of the resultant R and their singularities
Let us start with some general features of the roots u i pzq of R: -The polynomial Rpu, zq being symmetric in u, z, its roots u i pzq or z i puq are built by the same function: z This means that their graph is symmetric with respect to the first diagonal, see Fig. 10 below.
-Within the full domain q ď 0, there are either 0, two or four roots u i pzq P r0, 1s for z P r0, 1s. When z varies in p0, 1q, this number may change: either some of these roots may evade the interval r0, 1s through one of its end points 0 or 1, but they always do it pairwise; or a pair of real roots "pops out" of the complex plane or disappears into it, but this may occur only when they coalesce. In both cases, the discriminant ∆ R of R with respect to u vanishes. We compute where ∆ p1q R pz, p, qq is a fairly cumbersome polynomial of degree 8 in z that we refrain from writing here 2 . The vanishing of the first factor does not occur for q ă 0 and p ą´4. In the following, we denote the ordered roots z s of ∆ R belonging to r0, 1s as It turns out there are up to five roots z s of ∆ p1q R pz, p, qq in the open interval s0, 1r, hence seven in r0, 1s. Some of these roots z s may be irrelevant, in the sense that they are associated with the merging of irrelevant roots u i pzq of R, i.e., roots that do not satisfy u i P r0, 1s.
At some particular values of pp, qq, the number of z s roots may change. Either some z s evade the interval r0, 1s or enter it, through 0 or 1: computing ∆ p1q R p0, p, qq and ∆ p1q R p1, p, qq, one finds that this happens along the lines p˘q`1 " 0 and q " 0 which are the dashed lines in Fig. 6. Or two roots of ∆ p1q R pz, p, qq coalesce, and this occurs for values of pp, qq that are roots of the discriminant ∆ p2q R of ∆ p1q R pz, p, qq with respect to z, where T pp, qq is a horrendous polynomial of degree 21 in p and 18 in q 2 . The relevant roots of ∆ p2q R (in fact of T ) for our discussion define the "horned" (red) curve in the upper right part of Fig. 7. The cusp of that curve occurs at pp c , q c q " p´1.37657¨¨¨,´0.234765¨¨¨q. Note that this p c is a root of the "third generation discriminant" ∆ p3q R ppq :" ∆ T , in fact of its factor p23 p´p 2`3 p 3`p4 q. The horned curve intersects the q-axis at p " p 0 :"´1.21891¨¨¨, a root of 1328`1325p`171p 2´1 7p 3`p4 , which is a factor of T pp, 0q. Also, it intersects the dotted line p`q`1 " 0 at p "´0.910988¨¨¨, q "´0.089012¨¨¨; it intersects the dashed line p´q`1 " 0 at p "´1.14617¨¨¨, q "´0.146174¨¨¨.
A detailed analysis shows that one has to distinguish six regions in the domain of pp, qq, q ď 0, see Fig. 8. These regions differ by the subset of relevant values z s . Note that we have to carry out the z-integration of (33) in each interval pz s i , z s i`1 q, after one another, because the integrand is singular at each z s j , as we discuss in the next subsection. The properties of these six regions are summarized in Table 1; the pattern of z s when q varies while p is fixed at some value are displayed in Fig. 9; and the various scenarii for the roots u i pzq, which describe several branches of a closed curve in the pz, uq plane, are illustrated in Fig. 10, where colors refer to points of Fig. 7. R as a function of q: (a) for p "´3.5; (b) for p "´1.5; (c) for p "´1.25, for q min ppq ď q ď 0; (d) zoom of the latter on´.25 ď q ď 0. The cusps at q "´0.179503 and q "´0.0868984 lie on the boundary of the horned domain. In each case, the largest z s is irrelevant. Table 1. Pattern of roots z s of ∆ R and of roots u i pzq of R in the various regions For example, in region II, (shaded triangle X horned region), there are four roots z s j , j " 1,¨¨¨, 4, of ∆ p1q R but only the first three are relevant, and four relevant roots for ∆ R , namely z s 0 " 0, z s 1 , z s 2 , z s 3 : * for 0 ď z ď z s 1 , there are two roots 0 ď u 1 ď u 2 ď 1; * at z " z s 1 , a new pair of roots pu 3 , u 4 q pops out of the complex plane; * for z s 1 ă z ă z s 2 , we have four roots 0 ď u 1 ď u 3 ď u 4 ď u 2 ă 1; * at z " z s 2 , the pair pu 1 , u 3 q merges and disappears into the complex plane; * for z s 2 ď z ď z s 3 , we are left with two roots u 4 ď u 2 , which merge at z " z s 3 and disappear in the complex plane; * for z ą z s 3 , there is no root in the interval u P r0, 1s. The z-integration must be carried out separately on the three intervals p0, z s 1 q, pz s 1 , z s 2 q, and pz s 2 , z s 3 q.

The integrand ϕpzq
Consider the integrand in (33) It has both integrable and non-integrable singularities, the latter where the integral diverges. Typical plots of ϕpzq in the various regions are displayed in Fig. 11.  Figure 11: Typical plots of ϕpzq in regions I, V or VI (left) or II, III, IV (right). In the latter, the middle and right intervals have been dilated for clarity. The discontinuity of ϕpzq at z s i`1 , z s i`2 is due to the contribution of two new roots u 3 and u 4 in that interval.
The singularities of ϕ as a function of z come either from singularities of u i pzq, or from zeros of the denominator |R 1 u pu i q|. Both cases are associated with the merging of roots u i pzq of R, which occurs at some relevant root z s of its discriminant ∆ R . The singularity of u i in the numerator is, however, at worst of square root type and gives rise to no divergence of the integral. We thus concentrate on the possible vanishing of the denominator R 1 u . If we write the polynomial R in a factorized form, R " c ś j pu´u j pzqq, its derivative at u i reads R 1 u pu i q " c ś j‰i pu i pzq´u j pzqq, and vanishes when u i coalesces with some u j , thus at some value z " z s , a root of the discriminant ∆ R .
-Either the pair of roots pu i , u j q belonging to r0, 1s emerges from or disappears into the complex plane at z s with a square root behaviour, u i,j pzq " u i,j pz s q˘1 2 a|z´z s | 1{2`¨¨¨, hence |u i pzq´u j pzq| " a|z´z s | 1{2`. ... Graphically, it manifests itself as a smooth curvature of the "portraits" of u i pzq and u j pzq at the points of vertical tangent in Fig. 10. This is what happens at z " z s 0 " 0 above the dashed line; and generically at the various relevant roots z s of ∆ R . At such a point, the singularity of ϕpzq is integrable.
-Or the two roots u i , u j cross at a finite angle at z s : u i pzq´u j pzq " bpz´z s q`opz´z s q with a finite coefficient b. This is what happens along the lines p´q`1 " 0, or q " 0, and translates graphically into an angular point in Fig. 10, see for example cases (a),(d),(g). At such a point, the integral of ϕpzq diverges "logarithmically" at z " z s , which supposes the introduction of some cut-off that measures the departure of pp, qq from the singular point. This explains the growths of the PDF observed in Fig. 6 along the lines q " 0, p´q`1 " 0 and (by symmetry q Ø´q) p`q`1 " 0, as we discuss in the next subsection.
-Or at exceptional points, the difference u i pzq´u j pzq may vanish faster at z s . This is what happens at the point p´1, 0q, where it vanishes as |z s´z | 3{2 . Graphically, the two curves u i pzq and u j pzq form a cusp, see for instance Fig. 10.(f). In that case, the integral of ϕpzq diverges as an inverse power of the cut-off, see below.
But there is another source of divergence of ρ. The function ϕ itself may diverge as pp, qq approaches a singular point. This is what happens at the three corners p´4, 0q, p´3,´2q, p0, 0q of the domain, where we shall see that R 1 u pu i q vanishes for all z in the integration interval.

The PDF, plots and divergences
We are now in position to draw the plot of the PDF ρpp, qq resulting from the integration in (33) for q ď 0, supplemented by its mirror image by q Ñ´q, and to compare it with the histogram obtained by a simulation with 10 6 points, see Fig. 12 and 13. An important check consists in comparing the probability of occurrence of pp, qq in a finite domain computed by integrating the PDF ρpp, qq over that domain to that estimated from a big  sample of "random events". For example, Pp´3.6 ď p ď´3.5, q ď 0qˇˇc omputed " 0.04496 while the estimate from a sample of size 10 6 gives 0.044886. As anticipated, the computed ρ exhibits singularities along the lines q " 0 and p˘q`1 " 0. Note that the computation of ρ is carried out point by point on a grid of mesh 10´2 in the pp, qqplane, cutting off the vicinity of the singular lines, while the histogram uses bins of width 0.02 throughout the domain. This explains the slight difference of appearance of the singularities.
By a long (and fairly tedious) case by case analysis, we may assert that the singularities are logarithmic in the approach of generic points of the singular lines q " 0; p´q`1 " 0 for q ď 0; and p`q`1 " 0 for q ě 0. At the end points and intersection of these lines, i.e., at the corners of the pp, qq-domain, as well as the point pp, qq " p´1, 0q, the divergence is stronger, as an inverse power. This is summarized in the following Table, which gathers results obtained in the detailed discussion of the next subsection. The reader will also find in that subsection numerical verifications of the asserted divergences.
In most cases, we proceed as follows: as pp, qq approaches a singular point, some z s approaches a limiting value z s˚w hile the common value of a pair of coinciding roots, say, u s " u 1 pz s q " u 2 pz s q approaches u s˚" u 1,2 pz s˚q . Series expansions of z s and u s in powers of a "distance" ! 1 to the singularity may be computed. On the other hand, the roots u 1 pzq and u 2 pzq as well as the denominator R 1 u of ϕpzq in (33) may be expanded in powers of ζ " a |z´z s |. Finally, a double series expansion in powers of and ζ is obtained for R 1 u pu 1,2 pzqq, which upon integration in the vicinity of z s , yields the singular contribution to ρ of z s˚. This program is carried out in detail in the next subsection. Three particular singular points are treated separately.

Position of singularity
Approach to singularity Divergent part of ρ and pz s˚, u s˚q pp, qq " p´3,´2q pp`3q " κpq`2q Ñ 0 ρ div " C 1 pκq{pq`2q 1 2 z s˚" u s˚" 0´1 2 ď κ ď 1 Table 2. Position and expression of the singularities of ρpp, qq in the q ă 0 part of the domain of Fig. 6. The expressions of the coefficient functions Cpκq etc are given in the next subsection.
Finally, note the PDF ρpp, qq does not vanish along the boundaries of the Horn domain in the pp, qq-plane. In particular, on the lower or upper sides of the domain, i.e., along the arcs of the cubic q "¯2p´p{3q 3{2 , ρpp, qq has a finite limit. This is in no contradiction with the expected vanishing of the PDF ppγ 1 , γ 2 q on the left boundaries γ 1 " γ 2 and γ 2 " γ 3 of the Horn domain in the pγ 1 , γ 2 q-plane, because of (24) and of the vanishing of the Vandermonde determinant ∆ along those curves.

Analysis of the singularities
In this subsection, we proceed to a detailed -and lengthy-case-by-case analysis of the divergent singularities of ρ.
A preliminary observation is that, due to the u Ø z symmetry of our particular case, the u ă z and u ą z sectors will contribute equally: This will be apparent in the following.
As explained in the previous subsection, we proceed heuristically, making appropriate Taylor expansions close to the singularities. We do it in detail in the first case (singularity along the dashed line), and are then more sketchy.
1. Along the dashed line p`1´q " 0, the integrand ϕ has two non-integrable singularities: ϕ " 1{z for z Ñ z s 0 " 0, corresponding to u 1,2 pzq Ñ u s 0 " pp`3q{2, and, by symmetry between u and z, ϕ " 1{pz s´z q for z Ñ z s " pp`3q{2. Thus the integral diverges and the PDF is infinite along the line.
• For q " p`1` , (i.e., close to and above the line), z s remains equal to zero, and we determine the common value of the two roots at that point by plugging a series expansion of the form u s " p`3 2`α `¨¨¨in the equation Rpu s , z s " 0q " 0, whence For z close to z s 0 " 0, we write z " ζ 2 , approximate u 1,2 pzq " u s`β1,2 ζ`γ 1,2 ζ 2`O pζ 3 q, and determine the coefficients of that expansion by plugging it again in the equation Rpu 1,2 pzq, z " ζ 2 q " 0. β 1 (resp. β 2 ) is the negative (positive) root of an equation which, for q " p`1` reduces to and the coefficient γ is given by • We then expand the denominator R 1 u of (37), for z close to z s 0 " 0 and small and of order ζ 2 , as • and we finally derive the divergence of ş zs 0 dzϕ at the lower end point 0 as pp`1qpp`3q | log | .
As explained above, the divergence of the integral at its other end point, obtained by the symmetry u Ø z, contributes the same amount.
This formula is well verified on numerical data, see Fig. 14. Note that the logarithmic behavior is enhanced in the approach to p "´1, q " 0 and to p "´3, q "´2, at the southern end of the plot. This will be reconsidered in detail in items 4 and 7 below.
One also checks that the same formula applies to the approach to the dashed line from below ( ă 0). The relevant expressions are ith β ą 0 and γ 1,2 as given in (38) and (39), and the same expression for R 1 u as in (40): and hence the same divergence as above the line, see Fig. 14 for an illustration at p "´2.5.
2. By a similar discussion, one finds that when q Ñ 0 with´4 ď p ď p 0 "´1.21891, (i.e., in region I), the singularity at z s 0 " 0 is integrable while that at z s 1 Ñ 1 gives rise to a divergence of the integral. We write for short z s " z s 1 . For |q| small, z s " 1`2q{p4`pq`Opq 2 q , u s " u 1 pz s q " u 2 pz s q " 1´8q{ppp4`pqq`Opq 2 q ; as z Ñ z s , we write z " z s´ζ 2 , and the two roots u 1,2 « u s¯β1 ζ´γ 1,2 ζ 2 , for some computable coefficients β 1 , γ 1,2 , so that R 1 u pu 1,2 , z s´ζ 2 q «¯32|p|p ? 4`p ζ`?´2qqζ, whence a divergence of the integral ş zs dzϕpzq "  Table 1), but the singularities of R 1 u at the points z s 0 " 0, z s 1 and z s 2 are of inverse square root type, hence integrable, and only the linear vanishing of R 1 u at z s 3 matters. Hence ρpp, qqˇˇqÑ0 See Fig. 15 for comparison with numerical data.
As p Ñ´1, however, the singularities at z s 0 " 0 and z s 2 Ñ 1 become sharper and sharper, resulting in a stronger divergence at z " 1, see below item 4.
Thus as above in regions I and II, for q « 0, ρ " C log |q|, but with a larger value of the coefficient, C " 1 2π 2 |p| pp1`pq´1 2`p 4`pq´1 2 q. The agreement between numerical results and that coefficient C is illustrated in Fig. 16 for p "´0.8. It deteriorates at small values of |q| where the convergence of the integral is bad. 4. At p "´1, q " 0, one can see that, as z Ñ 0, u 1 and u 2 approach 1 with the same slope, so that we have now |u 1´u2 | " αz 3{2 , causing a strong divergence of ρ. More precisely, for ρ -ρdiv Figure 16: ρ´ρ div for p "´0.8, q "´3´j´3, plotted against j " 1,¨¨¨, 18.
p "´1 and q small, z s 1 "´q`Opq 2 q and if z´z s 1 " ζ 2 Ñ 0 , one finds whence a contribution to the divergent part of ş . The coalescence of roots u 1 and u 3 towards u s 4 « 0 at z s 4 « 1 gives rise to the same divergence (by the u´z symmetry once again), while there is a weaker (logarithmic) divergence coming from u 2 and u 4 merging to u s 3 « 1 as z Ñ z s 3 « 1 (see Fig. 10 f). In total, we have up to a subdominant log |q| term. See Fig. 17-left for a numerical plot of ρ{ρ div converging to 1.
Right: Plot of ρ{ρ div as a function of p, in the approach of p´4, 0q along the line p`q{κ "´4 for three values of κ.  Figure 18: (Left) Plot of ρpp, qq{ρ div in the approach of p0, 0q along the cubic κp 3`2 7q 2 " 0 for κ " 1, 2, 3 (in region VI). Convergence of the integral deteriorates for small values of p.
(Right) Plot of ρpp, qq{ρ div in the approach of p´3,´2q along the line p´qκ "´3`2κ for five values of κ.
whence a divergence of ρ as |q|´1 2 in very good agreement with numerical data, see Fig. 17-right.
6. Divergence at p " q " 0. We let pp, qq approach p0, 0q in region VI, for example along the cubic κp 3`2 7q 2 " 0, with κ ă 4; then the two end points z s 2 and z s 3 of the integral go to 1 as p "´ 2 goes to 0, i.e., z s i « 1´α z,i 2`O p 3 q where α z,i , i " 2, 3, are the second and third largest roots of 27α z p1´α z q 2´κ . Both z and u are thus confined in an interval of size 2 near 1, and solving the equation R " 0 in the rescale variable z " 1´ζ 2 and plugging into R 1 u , one finds that the latter has a limiting shape described by the elliptic curve This behavior is again well supported by numerical calculations, see Fig. 18-left.
7. Divergence at p "´3, q "´2. If one approaches that corner of the domain along lines p "´3`κ , q "´2` with´1 2 ă κ ă 1 so as to remain in region I, one finds that z s 0 " 0 and z s 1 " 1 6 p1`2κq to the lowest order in , so that both z and u remain small of order . Solving the equation R " 0 to order 3 , one finds that u 1 2 pzq " z s 1´z˘2 9´1 2 p1´κq`zpz s 1´z q˘¯1 2 so that |R 1 u pu 1,2 q| « 32 a 3p1´κq`zpz s 1´z q˘1 2 1 2 , and the z-integration may be carried out, leading to This is corroborated by the numerical calculation at various values of κ, see Fig. 18-right.
The alert reader may wonder why the singularity along the line p`q`1 " 0 of the upper half-plane, (a reflection of the singularity along the dashed line of the q ă 0 half-plane) does not manifest itself along the dotted line of the lower half-plane. The reason is that, in that lower half-plane, the two z s that merge there are in fact irrelevant for q ă 0.

Zonal polynomials
In the Hermitian case, it is known that the Horn problem discussed so far has a discrete counterpart, involving Littlewood-Richardson multiplicities, and may be regarded as a semi-classical limit of the latter. The PDF ppγ|α, βq, or rather the "volume function" J equal to the latter up to a Vandermonde factor, measures the distribution of (rescaled) Littlewood-Richardson multiplicities, i.e., structure constants of Schur polynomials, in the large scale limit, see [5,16]. In the real symmetric case, one expects similarly the PDF, or rather some "volume like function" J proportional to it to measure, at least in the generic case, the distribution of (rescaled) "zonal multiplicities" i.e., appropriate structure constants of zonal polynomials, see [9]. This motivates the discussion of the present section, where we pay special attention to the normalization and specializations of the Jack and zonal polynomials. We shall in particular use SUpnq reduction 5 of zonal (or Jack) polynomials and introduce a notion of SUpnq zonal characters very similar to the usual Weyl characters.

Jack polynomials and their normalizations
Zonal polynomials can be defined in many ways and we refer the reader to the abundant literature (see for instance [17], [18]). One possible approach is to start from Jack polynomials (themselves a particular case of the larger family called Macdonald polynomials). Sentence re-written A Jack polynomial with n variables is labelled by an integer partition κ and a real parameter α. When one specializes the value of α, Jack polynomials, in turn, give rise to various interesting families, in particular the Schur polynomials (case α " 1), the zonal polynomials (case α " 2), and the quaternionic polynomials (case α " 1{2).
Actually there are three variants of the Jack polynomials, denoted J α X with X " P, C, J 6 differing by an overall normalization (an overall α-dependent and partition-dependent numerical factor): for example, one writes J J " c P J J P , etc. When α " 2 one has therefore also three kinds of zonal polynomials respectively denoted Z P , Z J and Z C . When studying zonal or Jack polynomials, many authors -in particular in old papers -use the James normalization (polynomials Z J ) without saying so explicitly. As we shall see, the most interesting family, for us, is the family of the zonal polynomials (and also the Jack polynomials) defined with the P normalization, the reason, that will be discussed below, is that this normalization is compatible with SUpnq-reduction and with the conjugation of irreducible representations (irreps) of SUpnq -the latter being described by integer partitions with at most n´1 parts. When expanded in terms of monomial symmetric polynomials, the J normalization of the Jack polynomial defined by the partition κ makes the coefficient of the lowest order monomial r1 n s equal to n!, whereas, using the same expansion, the P normalization makes the coefficient of the monomial relative to the highest partition (i.e., κ) equal to 1. For a given partition κ, the normalization factor c P J is the lower α-hook coefficient of κ. Note: Zonal polynomials, with an un-specified normalization, were denoted Zpκqpxq in (5).

Packages
Zonal polynomials and, more generally, Jack polynomials (variables are called x j ), are usually written in terms of monomial symmetric functions, or in terms of power sums, not very often in terms of the variables x j themselves because this would take too much space. To the authors' knowledge there are very few computer algebra packages devoted to the manipulation of those polynomials; we should certainly mention [23], written for Mapple (that we did not use), and the small package [24] written for Mathematica (but it is slow, unstable (division by 0), and uses an obsolete version of the language). For those reasons we developed our own, using Mathematica: the definition chosen for Jack polynomials uses a recurrence algorithm in terms of skew Young diagrams and a modified Pieri's formula described by Macdonald in [18], see also [25]; our code, which also contains commands to convert Jack, Schur and zonal polynomials to several other basis (elementary symmetric polynomials, power sums, monomial sums, complete sums), and commands for calculating structure constants in each basis, is freely available on the web site [14]; the same package contains commands giving the coefficients ZpκqpIq and cpκq that appear in formula (5), with various normalization choices.

Structure constants
Zonal polynomials form a basis of the space of the ring of symmetric polynomials in n variables. Structure constants in this basis are only rationals (by way of contrast, the coefficients of Schur polynomials in the expansion of a product of two Schur polynomials are non-negative integers). For illustration, let us consider the zonal polynomial(s) for the extended 7 partition t2, 1, 0u (i.e., three variables x 1 , x 2 , x 3 ), and the decomposition of its square, using the three standard normalizations; we also give the decomposition obtained for the square of the Schur polynomial spt2, 1, 0uq. ZC pt4, 2, 0uq spt2, 1, 0uq 2 " spt2, 2, 2uq`2 spt3, 2, 1uq`spt3, 3, 0uq`spt4, 1, 1uq`spt4, 2, 0uq .

A particular feature of structure constants in the Z P basis
We pause here to notice that the coefficients of Z P pt3, 3, 0uq and of Z P pt4, 1, 1uq are the same (both equal to 4{3), with the same remark for the coefficients of spt3, 3, 0uq and of spt4, 1, 1uq, which are both equal to 1. This is not so for Z J and Z C .
More generally we observe (the proof of this conjecture is left to the reader) that the following property holds in the zonal P case 9 : If λ is a self-conjugate irreducible representation of SUpnq described by a Young diagram of shape χ (an integer partition, or an extended partition of length n), and if χ 1 and χ 2 are two partitions appearing in the decomposition of the square of the zonal polynomial Z P pχq in the Z P basis that give rise, after SUpnq reduction, to complex conjugate representations, the coefficients (structure constants) of Z P pχ 1 q and of Z P pχ 2 q are equal. Actually, the same property seems to hold for all values of the Jack parameter α, when using the P normalization. It does not hold for the normalizations J and C of zonal polynomials.
Remember that the notion of complex conjugation on SU(n) irreps can be described in purely combinatorial terms: if κ is the integer partition describing some irrep of SU(n), its length (number of parts) obeys ιpκq ă n; then one obtains the partition describing the complex conjugate representation by taking the complement of (the Young diagram of) κ in a rectangle which is κp1q units wide (κp1q being the largest part of κ) and pιpκq`1q units deep.

From zonal polynomials to SUpnq zonal characters
Although we have in mind applications to the zonal case α " 2, or to the quaternionic (zonal) case α " 1{2, most of our considerations, in the section that follows, apply to arbitrary values α of the Jack parameter.

SUpnq-zonal characters
An irrep of SUpnq, is characterized by its highest weight (hw) λ. Its components in the basis of fundamental weights (Dynkin labels) are denoted rλ 1 , . . . , λ n´1 s. When considering irreps of Upnq one adds a last index λ n ; here we only consider the case SUpnq, but it is often handy to keep this last index, while setting λ n " 0. One can also characterize the same irrep λ by the Young diagram defined by the extended partition α " pλq with components i pλq " ř n j"i λ j , i " 1,¨¨¨, n, obeying the constraint α i ě α i`1 for all i. Conversely, given a partition δ of length n, extended or not, one obtains a highest weight λ for SUpnq by setting λ i " δ i´δi`1 ; such a partition δ differs from α " pλq by a constant shift. We denote by N ν λµ the multiplicity 10 of the irrep ν in the tensor product of the irreps of SUpnq defined by λ and µ.
Given a dominant weight λ of SUpnq, i.e., a non-negative integer combination of the fundamental weights, call pλq its associated partition of length n (i.e., pλq n " 0) and take the Jack-P polynomial J α P p pλqqpx 1 , . . . , x n q determined by the partition pλq. We then consider the following Laurent polynomial in the variables y 1 , . . . , y n´1 : If α " 1, J α P is a Schur polynomial and the previous Laurent polynomial is recognized as the Weyl character χpλq of the irrep λ, for the Lie group SUpnq. If α " 2, i.e., when J α P is a zonal polynomial Z P (with the normalization P ), we introduce, by analogy, and for lack of a better name, the following notation and definition 11 : 9 We remind the reader that Schur polynomials can be obtained from Jack polynomials, with the P normalization, just by setting α " 1 (no pre-factors).
10 N ν λµ is sometimes called the Littlewood-Richardson (LR) multiplicity, although, strictly speaking, the latter refers to the coefficient of pνq in the decomposition in the Schur basis of the product of two Schur polynomials respectively defined by the partitions pλq and pµq. This decomposition often contains terms labelled by integer partitions of length larger than n, therefore not contributing to the tensor product of SUpnq representations-a Young diagram of SUpnq cannot have more than n´1 lines. 11 Another notion of "zonal character" can be found in the literature, [26], but it is related to the symmetric group, not to irreducible representations of SUpnq. It differs from the notion that we consider here. Definition 1. The zonal character χ Z pλq of SUpnq associated with the dominant weight λ is defined as the Laurent polynomial χ Z pλqpy 1 , . . . , y n´1 q " Z P p pλqqpy 1 , y 2 y 1 , . . . , y j y j´1 , . . . , y n´1 y n´2 , 1 y n´1 q Now, moving to the Lie algebra supnq -or, equivalently, to trigonometric characters, we start from the same Z P polynomial expressed in terms of x j variables but this time perform the following transformations on its arguments: x i Ñ e ipa i´1 n ř n j a j q , then a j Ñ a 1´ř j´1 i"1 u i . The result is a trigonometrical expression in the variables u j , that we call the Lie algebra zonal character of supnq associated with the hw λ, or the trigonometric zonal character of SUpnq associated with the hw λ.
Let us give one example. Take n " 3 and λ " r1, 1s. The associated (extended) partition is pλq " t2, 1, 0u. The Jack polynomial J α P pt2, 1, 0uq in terms of the variables x 1 , x 2 , x 3 , the associated Laurent polynomial, and its trigonometric version are given below. The Schur polynomial spt2, 1, 0uq and the Zonal-P polynomial Z P pt2, 1, 0uq are obtained from the first expression by setting respectively α " 1 and α " 2. The corresponding SUpnq and supnq zonal characters are obtained from the last two expressions by setting α " 2 α`2`c os u 1`c os u 2`c os pu 1`u2 qṪ aking α " 1 in the second expression, one recognizes the Weyl character of the adjoint representation of SU(3), the powers (positive or negative) of the y j being, as usual, the components of the weights of the weight system of this representaton in the basis of fundamental weights. What happens in the zonal case, and more generally when α ‰ 1 is that the "multiplicities" of the weights are no longer integers. For this particular irrep, only the multiplicity of the weight at the origin of the weight system is modified by α, its value being 2 in the usual (Schur) case but 3{2 in the zonal case. Notice that the trigonometric expression is real -it is so because the hw r1, 1s is self-conjugate, otherwise the obtained expression would be complex. The arguments being specified (partitions or Dynkin labels), we shall denote χ SUpnq α , in the cases α " 2 and α " 1{2, by χ Z and χ Q .

Structure constants for SU(n)-zonal characters
Given two SUpnq irreps, there are many ways to obtain the decomposition of their tensor products into a sum of irreps. The honeycomb technique, for instance, is very fast 12 but it is not available in the zonal case that we consider. However, we can replace the multiplication of the associated SUpnq Weyl characters by the multiplication of the associated SUpnq zonal characters as defined above, which amounts to use the structure constants for the appropriate product of zonal polynomials. Let us illustrate this with our favorite example, the square of r1, 1s. From the already given decomposition of the square of Z P pt2, 1, 0uq we obtain immediately: The same decomposition can be obtained by using Laurent polynomials since the associated SU(3) zonal characters are as follows (the reader can then check that the previous equality holds): he previous result -and more generally any decomposition of a product of such characters -can be checked by using a concept of dimension. The dimension of an irreducible representation is the value taken by the associated Weyl SUpnq character at y j " 1 (or the value taken by the supnq character at u j " 0), for all j. In the same way one can define a "zonal dimension" for an irrep of hw λ as the value taken by the SUpnq zonal character χpλq for y j " 1 (or the value taken by the supnq character for u j " 0). This is a slight terminological abuse since the obtained number is not an integer in general, but this dimension function is obviously compatible with addition and multiplication, as it should. In the case 13 of SU(3), we conjecture that this expression looks very similar to the standard SU(3) dimension dimprλ 1 , λ 2 sq " p1`λ 1 qp1`λ 2 qp2`λ 1`λ2 q 2 " Γpλ 1`2 qΓpλ 2`2 qΓpλ 1`λ2`3 q 2Γpλ 1`1 qΓpλ 2`1 qΓpλ 1`λ2`2 q . This zonal dimension is (non-surprisingly) related to the normalization coefficient Z p pIq that enters (5), and was introduced by A.T. James in [12]: Z pλq pIq c P J p pλq, α " 2q It may be interesting to notice that both the numerator and the denominator of this formula are not invariant under a global shift (translation of the partition pλq by an arbitrary integer), but their ratio is invariant -this can be interpreted as a kind of "gauge freedom" in the writing of the SUpnq highest weight λ as a partition.

Back to the PDF (symmetric case) and to the "volume function"
In the Hermitian case it is known that one may associate with a given admissible triple pλ, µ, νq a convex polytope H ν λµ , the "polytope of honeycombs", in a d ď pn´1qpn´2q{2-dimensional space [3]. As recalled above, it is known that the function called J in [16], which differs from the PDF ppγq mainly by a Vandermonde factor ∆, measures the volume of H ν λµ and that it is also a good approximation of the LR multiplicity N ν λµ of that triple 14 . More precisely J is equal to the highest degree coefficient of the stretching (or LR) polynomial that gives the multiplicity when the triple pλ, µ, νq is scaled by a factor s, i.e., J is also the dominant coefficient of the Ehrhart polynomial of the polytope H ν λµ . Since this multiplicity is known to be given by the number of integral points inside the polytope [3], this property is just expressing that in the large s limit, a semi-classical picture approximates well this number of points by the volume 15 of the polytope.
In the symmetric case multiplicities are not integral, there are no honeycombs (at least the concept was not (yet?) generalized to cover this case), no polytope H ν λµ , and no volume function either. However, as already mentioned, one expects the PDF ppγq, or rather the "volume like function" J " ppγq{∆ " ρ, (see (17,24)), to measure, at least in the generic case where J does not vanish, the behavior of "zonal multiplicities" (i.e., appropriate zonal structure constants) under scaling.
A claim going in that direction was actually already made in [9]. Some of the tools developed in the previous subsection could certainly be developed further and we hope to return to this problem someday but we are happy, in the present paper, to show "experimentally" that the overall features of the PDF ρ computed in section 4 are consistent with the values obtained for zonal multiplicities, when the argument (written as a highest weight), is scaled. Remember that the list of eigenvalues p1, 0,´1q chosen for the example studied in sections 2, 3 and 4 differs from the partition t2, 1, 0u (aka r1, 1s if reinterpreted in terms of SU(3) highest weights) only by a constant shift 16 ; we are therefore led to consider the behavior of the zonal structure constants that appear in the reduction of the square of st2, 1, 0u, with a scaling factor s " 1, 2, . . ., with the plot of the function ρ in terms of Dynkin labels. Using [14] we could perform exact calculations of multiplicities, i.e., obtain the decomposition of the square of χ Z psr1, 1sq up to the value s " 6 of the scaling factor 17 . The same considerations and calculations extend to the quaternionic case, where we compare multiplicities in the decomposition of χ Q pr6, 6sq 2 with the function J 3 computed in sect. 2.
In Fig. 19, we compare the multiplicities obtained for the decomposition of χ Z pr6, 6sq 2 , χpr6, 6sq 2 and χ Q pr6, 6sq 2 with the plot of the volume functions ρ or J 3 . We conclude that already with s " 6, the classical limit provided by the "volume" approximates very well the distribution of multiplicities.

Conclusion
To summarize: -we have reproduced the main features of the Horn problem for symmetric matrices and understood the analytic origin of the singularities, at least for n " 3, and in detail for the particular case of A " B " J z ; -we have confirmed, at least in that particular case, that the divergences of the PDF occur on the same locus of non-analyticities as in the Hermitian and quaternionic cases; -we have also confirmed numerically the connection between the "volume functions" and the (asymptotic) distribution of multiplicities in the product of zonal/Schur/quaternionic polynomials. This leaves, however, open several issues and room for further progress: -a more synthetic and general discussion of the singularities in the symmetric case would clearly be desirable. Can one understand their origin from a geometric point of view and assert a priori their location and nature without detailed calculations? Beside the divergences analyzed in the present paper, are there other non-analyticities? -what happens for higher n and/or for generic β (or α " 2{β)? The methods developed recently in [29] should be helpful in that respect.
The points we find most challenging are the following: -There is an enhancement of particular eigenvalues in the Horn spectrum of real symmetric matrices, due to the divergences of the PDF. Is this enhancement observable in some physical process? -The discussion of sect. 5 has pointed to an analogue of the volume function for real symmetric or quaternionic matrices: is there an underlying geometric interpretation to this "volume"? is there a geometric object generalizing the polytope H ν λµ of the Hermitian case, whose volume is com-puted there? on a representation theoretic side, what is the origin of the enhancement of certain multiplicities ?
We leave these questions to the sagacity of our readers. . .