On Horn’s Problem and Its Volume Function

Abstract

We consider an extended version of Horn’s problem: given two orbits ${{\mathcal {O}}}_\alpha$ and ${{\mathcal {O}}}_\beta$ of a linear representation of a compact Lie group, let $A\in {\mathcal O}_\alpha$, $B\in {{\mathcal {O}}}_\beta$ be independent and invariantly distributed random elements of the two orbits. The problem is to describe the probability distribution of the orbit of the sum $A+B$. We study in particular the familiar case of coadjoint orbits, and also the orbits of self-adjoint real, complex and quaternionic matrices under the conjugation actions of $\mathrm{SO}(n)$, $\mathrm{SU}(n)$ and $\mathrm {USp}(n)$ respectively. The probability density can be expressed in terms of a function that we call the volume function. In this paper, (i) we relate this function to the symplectic or Riemannian geometry of the orbits, depending on the case; (ii) we discuss its non-analyticities and possible vanishing; (iii) in the coadjoint case, we study its relation to tensor product multiplicities (generalized Littlewood–Richardson coefficients) and show that it computes the volume of a family of convex polytopes introduced by Berenstein and Zelevinsky. These considerations are illustrated by a detailed study of the volume function for the coadjoint orbits of $B_2=\mathfrak {so}(5)$.

Appendix: Covolumes

Let us sketch how the covolume $c_{{\mathfrak {g}}}$ may be determined for the lattice $\Lambda$ of integer points of $\mathrm {aff}(H_{\lambda \mu }^\nu )$, in the case that $\dim H_{\lambda \mu }^\nu = d_r$ so that $\mathrm {aff}(H_{\lambda \mu }^\nu ) = \mathrm {aff}(\mathrm {Part}_{{\mathfrak {g}}}(\sigma ))$, see (33). This affine span has codimension r in the Euclidean space $\mathbb {R}^{N_r}$ generated as the formal span of the positive roots, which are taken to form a canonical basis, i.e. $\langle \pmb {\alpha }_a, \pmb {\alpha }_b\rangle = \delta _{ab}$, $a,b=1,\cdots , N_r$. Since the covolume of $\Lambda$ is independent of the value of $\sigma$, we may take $\sigma =0$. Thus we consider the lattice of integer combinations of the positive roots $\pmb {\alpha }_a$ subject to the condition

$$\begin{aligned} \sum _{a=1}^{N_r} u_a \pmb {\alpha }_a =0\in {\mathfrak {t}}^*. \end{aligned}$$

(58)

Suppose that the first r of the $\pmb {\alpha }$’s are the simple roots, and denote by A the $(N_r-r) \times r$ matrix that expresses the non-simple roots in terms of the simple ones:

$$\begin{aligned} \pmb {\alpha }_a=\sum _{i=1}^r A_{ai}\pmb {\alpha }_i, \quad a=r+1,\ldots , N_r. \end{aligned}$$

Then eliminate the parameters $u_i$ for $i=1,\ldots ,r$ in the condition (58), using the relation $u_i= - \sum _{a=r+1}^{N_r} u_a A_{ai}$. The lattice is thus generated by the linear combinations $\sum _{a=r+1}^{N_r} u_a (-\sum A_{ai} \pmb {\alpha }_i +\pmb {\alpha }_a)$. Under the assumption that its fundamental domain is generated by the $N_r-r$ vectors $w_a= (-\sum A_{ai} \pmb {\alpha }_i +\pmb {\alpha }_a)$, we can compute the volume of this fundamental domain, i.e. the covolume, as follows. The Gram matrix of the vectors $w_a$ in the Euclidean space $\mathbb {R}^{N_r}$ reads

$$\begin{aligned} G_{ab}=\langle w_a, w_b\rangle = \delta _{ab}+ \sum _{i=1}^r A_{ai}A_{bi}\,. \end{aligned}$$

The covolume is then the square root of the determinant $\delta _r$ of G, $c_{{\mathfrak {g}}}=\delta _r^\frac{1}{2}$. We have computed these determinants for the classical algebras $A_r,\ B_r,\ C_r$ and $D_r$ up to $r=8$ and the values of $\delta _r$ listed in Table 1 are extrapolations of these numbers. The values for the exceptional algebras are also included in Table 1. A last observation is that for all the simple algebras, a single formula encompasses the expressions found in the table:

$$\begin{aligned} \delta _r= \frac{(h^\vee )^r}{\det C} \, \prod _{i=1}^r \frac{\langle \pmb {\theta }, \pmb {\theta } \rangle }{\langle \pmb {\alpha }_i,\pmb {\alpha }_i\rangle } \,, \end{aligned}$$

(59)

where C is the Cartan matrix, $h^\vee$ is the dual Coxeter number and the product runs over the simple roots.