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1. CJM Online first

Benaych-Georges, Florent; Cébron, Guillaume; Rochet, Jean
Fluctuation of matrix entries and application to outliers of elliptic matrices
For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in K}$ which is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type $\operatorname{Tr}(\mathbf{A}_k \mathbf{M})$, where the matrix $\mathbf{M}$ is deterministic (such random variables include for example the normalized matrix entries of the $\mathbf{A}_k$'s). A consequence is the asymptotic independence of the projection of the matrices $\mathbf{A}_k$ onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Form of the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other. These phenomena have already been observed with random matrices from the Single Ring Theorem.

Keywords:random matrix, Gaussian fluctuation, spiked model, elliptic random matrix, Weingarten calculus, Haar measure
Categories:60B20, 15B52, 60F05, 46L54

2. CJM Online first

Bremner, Murray; Dotsenko, Vladimir
Classification of regular parametrized one-relation operads
Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: \[ (a_1a_2)a_3=\sum_{\sigma\in S_3}x_\sigma\, a_{\sigma(1)}(a_{\sigma(2)}a_{\sigma(3)})\ ; \] such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{x_\sigma\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the relation $((a_1a_2)a_3)=0$, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

Keywords:parametrized one-relation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, Gröbner basis
Categories:18D50, 13B25, 13P10, 13P15, 15A54, 17-04, , , , , 17A30, 17A50, 20C30, 68W30

3. CJM 2016 (vol 68 pp. 395)

Garibaldi, Skip; Nakano, Daniel K.
Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given complex representation is symplectic or orthogonal has been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups
Categories:20G05, 11E39, 11E88, 15A63, 20G15

4. CJM 2015 (vol 69 pp. 408)

Klep, Igor; Špenko, Špela
Free Function Theory Through Matrix Invariants
This paper concerns free function theory. Free maps are free analogs of analytic functions in several complex variables, and are defined in terms of freely noncommuting variables. A function of $g$ noncommuting variables is a function on $g$-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps \emph{with involution} -- free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invariant-theoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involution-free counterparts.

Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomials
Categories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20

5. CJM 2015 (vol 67 pp. 961)

Abuaf, Roland; Boralevi, Ada
Orthogonal Bundles and Skew-Hamiltonian Matrices
Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space $M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial splitting on the general line, is smooth irreducible of dimension $(r-2)n-\binom{r}{2}$ for $r=n$ and $n \ge 4$, and $r=n-1$ and $n\ge 8$. We speculate that the result holds in greater generality.

Keywords:orthogonal vector bundles, moduli spaces, skew-Hamiltonian matrices
Categories:14J60, 15B99

6. CJM 2014 (vol 67 pp. 241)

Agler, Jim; McCarthy, John E.
Global Holomorphic Functions in Several Noncommuting Variables
We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.

Keywords:noncommutative analysis, free holomorphic functions

7. CJM 2013 (vol 65 pp. 1287)

Reihani, Kamran
$K$-theory of Furstenberg Transformation Group $C^*$-algebras
The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism
Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20

8. CJM 2012 (vol 65 pp. 1020)

Goulden, I. P.; Guay-Paquet, Mathieu; Novak, Jonathan
Monotone Hurwitz Numbers in Genus Zero
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

Keywords:Hurwitz numbers, matrix models, enumerative geometry
Categories:05A15, 14E20, 15B52

9. CJM 2011 (vol 63 pp. 1364)

Meinrenken, Eckhard
The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras
Let $\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}} \mathfrak{g}_i$ be an infinite-dimensional graded Lie algebra, with $\dim\mathfrak{g}_i<\infty$, equipped with a non-degenerate symmetric bilinear form $B$ of degree $0$. The quantum Weil algebra $\widehat{\mathcal{W}}\mathfrak{g}$ is a completion of the tensor product of the enveloping and Clifford algebras of $\mathfrak{g}$. Provided that the Kac-Peterson class of $\mathfrak{g}$ vanishes, one can construct a cubic Dirac operator $\mathcal{D}\in\widehat{\mathcal{W}}(\mathfrak{g})$, whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac-Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl-Kac character formulas for suitable ``equal rank'' Lie subalgebras of Kac-Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras.

Categories:22E65, 15A66

10. CJM 2010 (vol 63 pp. 413)

Konvalinka, Matjaž; Skandera, Mark
Generating Functions for Hecke Algebra Characters
Certain polynomials in $n^2$ variables that serve as generating functions for symmetric group characters are sometimes called ($S_n$) character immanants. We point out a close connection between the identities of Littlewood--Merris--Watkins and Goulden--Jackson, which relate $S_n$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra $H_n(q)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood--Merris--Watkins identities and selected Goulden--Jackson identities that relate $H_n(q)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis--L\^e--Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.

Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring
Categories:15A15, 20C08, 81R50

11. CJM 2010 (vol 63 pp. 3)

Banica, T.; Belinschi, S. T.; Capitaine, M.; Collins, B.
Free Bessel Laws
We introduce and study a remarkable family of real probability measures $\pi_{st}$ that we call free Bessel laws. These are related to the free Poisson law $\pi$ via the formulae $\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

Keywords:Poisson law, Bessel function, Wishart matrix, quantum group
Categories:46L54, 15A52, 16W30

12. CJM 2010 (vol 62 pp. 758)

Dolinar, Gregor; Kuzma, Bojan
General Preservers of Quasi-Commutativity
Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasi-commute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasi-commutativity in both directions.

Keywords:general preservers, matrix algebra, quasi-commutativity
Categories:15A04, 15A27, 06A99

13. CJM 2009 (vol 62 pp. 109)

Li, Chi-Kwong; Poon, Yiu-Tung
Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues
Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$. All possible inertia values, ranks, and multiple eigenvalues of $A + B$ are determined. Extension of the results to the sum of $k$ matrices with $k > 2$ and connections of the results to other subjects such as algebraic combinatorics are also discussed.

Keywords:complex Hermitian matrices, real symmetric matrices, inertia, rank, multiple eigenvalues
Categories:15A42, 15A57

14. CJM 2008 (vol 60 pp. 1149)

Petersen, Kathleen L.; Sinclair, Christopher D.
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle
We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree $N$ is naturally associated to a subset of $\R^{N-1}$. We calculate the volume of this set, prove the set is homeomorphic to the $N-1$ ball and that its isometry group is isomorphic to the dihedral group of order $2N$.

Categories:11C08, 28A75, 15A52, 54H10, 58D19

15. CJM 2008 (vol 60 pp. 1050)

Huang, Wen-ling; Semrl, Peter \v
Adjacency Preserving Maps on Hermitian Matrices
Hua's fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all $n\times n$ hermitian matrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization for maps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

Keywords:rank, adjacency preserving map, hermitian matrix, geometry of matrices
Categories:15A03, 15A04, 15A57, 15A99

16. CJM 2008 (vol 60 pp. 923)

Okoh, F.; Zorzitto, F.
Endomorphisms of Kronecker Modules Regulated by Quadratic Algebra Extensions of a Function Field
The Kronecker modules $\mathbb{V}(m,h,\alpha)$, where $m$ is a positive integer, $h$ is a height function, and $\alpha$ is a $K$-linear functional on the space $K(X)$ of rational functions in one variable $X$ over an algebraically closed field $K$, are models for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial $f$ in $K(X)[Y]$. When the endomorphism algebra of $\mathbb{V}(m,h,\alpha)$ is commutative and non-trivial, the regulator $f$ must be quadratic in $Y$. If $f$ has one repeated root in $K(X)$, the endomorphism algebra is the trivial extension $K\ltimes S$ for some vector space $S$. If $f$ has distinct roots in $K(X)$, then the endomorphisms form a structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those $\End\mathbb{V}(m,h,\alpha)$ that are domains have zero radical. In addition, each semi-local $\End\mathbb{V}(m,h,\alpha)$ must be either a trivial extension $K\ltimes S$ or the product $K\times K$.

Categories:16S50, 15A27

17. CJM 2008 (vol 60 pp. 520)

Chen, Chang-Pao; Huang, Hao-Wei; Shen, Chun-Yen
Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences
Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: $\ell_p$, $d(w,p)$, and $\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, Nörlund mean matrices, summability matrices, matrices with row decreasing
Categories:15A60, 40G05, 47A30, 47B37, 46B42

18. CJM 2007 (vol 59 pp. 1284)

Fukshansky, Lenny
On Effective Witt Decomposition and the Cartan--Dieudonn{é Theorem
Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical theorem of Witt states that the bilinear space $(Z,F)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan--Dieudonn{\'e} theorem. Namely, we show that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $\sigma$.

Keywords:quadratic form, heights
Categories:11E12, 15A63, 11G50

19. CJM 2007 (vol 59 pp. 488)

Bernardi, A.; Catalisano, M. V.; Gimigliano, A.; Idà, M.
Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties
We consider the $k$-osculating varieties $O_{k,n.d}$ to the (Veronese) $d$-uple embeddings of $\PP^n$. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes $Y\subset \PP^n$ to $O^s_{k,n,d}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

Categories:14N15, 15A69

20. CJM 2007 (vol 59 pp. 638)

MacDonald, Gordon W.
Distance from Idempotents to Nilpotents
We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of $n\times n$ matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.

Keywords:operator, matrix, nilpotent, idempotent, projection
Categories:47A15, 47D03, 15A30

21. CJM 2007 (vol 59 pp. 186)

Okoh, F.; Zorzitto, F.
Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields
Purely simple Kronecker modules ${\mathcal M}$, built from an algebraically closed field $K$, arise from a triplet $(m,h,\alpha)$ where $m$ is a positive integer, $h\colon\ktil\ar \{\infty,0,1,2,3,\dots\}$ is a height function, and $\alpha$ is a $K$-linear functional on the space $\krx$ of rational functions in one variable $X$. Every pair $(h,\alpha)$ comes with a polynomial $f$ in $K(X)[Y]$ called the regulator. When the module ${\mathcal M}$ admits non-trivial endomorphisms, $f$ must be linear or quadratic in $Y$. In that case ${\mathcal M}$ is purely simple if and only if $f$ is an irreducible quadratic. Then the $K$-algebra $\edm\cm$ embeds in the quadratic function field $\krx[Y]/(f)$. For some height functions $h$ of infinite support $I$, the search for a functional $\alpha$ for which $(h,\alpha)$ has regulator $0$ comes down to having functions $\eta\colon I\ar K$ such that no planar curve intersects the graph of $\eta$ on a cofinite subset. If $K$ has characterictic not $2$, and the triplet $(m,h,\alpha)$ gives a purely-simple Kronecker module ${\mathcal M}$ having non-trivial endomorphisms, then $h$ attains the value $\infty$ at least once on $\ktil$ and $h$ is finite-valued at least twice on $\ktil$. Conversely all these $h$ form part of such triplets. The proof of this result hinges on the fact that a rational function $r$ is a perfect square in $\krx$ if and only if $r$ is a perfect square in the completions of $\krx$ with respect to all of its valuations.

Keywords:Purely simple Kronecker module, regulating polynomial, Laurent expansions, endomorphism algebra
Categories:16S50, 15A27

22. CJM 2005 (vol 57 pp. 82)

Fallat, Shaun M.; Gekhtman, Michael I.
Jordan Structures of Totally Nonnegative Matrices
An $n \times n$ matrix is said to be totally nonnegative if every minor of $A$ is nonnegative. In this paper we completely characterize all possible Jordan canonical forms of irreducible totally nonnegative matrices. Our approach is mostly combinatorial and is based on the study of weighted planar diagrams associated with totally nonnegative matrices.

Keywords:totally nonnegative matrices, planar diagrams,, principal rank, Jordan canonical form
Categories:15A21, 15A48, 05C38

23. CJM 2004 (vol 56 pp. 776)

Lim, Yongdo
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices
We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold ${\mathrm{Sym}}(p,{\mathbb R})^{++}\times {\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in ${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$

Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform
Categories:15A48, 49R50, 15A18, 53C3

24. CJM 2004 (vol 56 pp. 134)

Li, Chi-Kwong; Sourour, Ahmed Ramzi
Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States
Every norm $\nu$ on $\mathbf{C}^n$ induces two norm numerical ranges on the algebra $M_n$ of all $n\times n$ complex matrices, the spatial numerical range $$ W(A)= \{x^*Ay : x, y \in \mathbf{C}^n,\nu^D(x) = \nu(y) = x^*y = 1\}, $$ where $\nu^D$ is the norm dual to $\nu$, and the algebra numerical range $$ V(A) = \{ f(A) : f \in \mathcal{S} \}, $$ where $\mathcal{S}$ is the set of states on the normed algebra $M_n$ under the operator norm induced by $\nu$. For a symmetric norm $\nu$, we identify all linear maps on $M_n$ that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, {\it i.e.}, linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\nu$ is not the $\ell_1$, $\ell_2$, or $\ell_\infty$ norms, then the linear maps that preserve either numerical range or either set of states are ``inner'', {\it i.e.}, of the form $A\mapsto Q^*AQ$, where $Q$ is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the $\ell_1$ and the $\ell_\infty$ norms, the results are quite different.

Keywords:Numerical range, numerical radius, state, isometry
Categories:15A60, 15A04, 47A12, 47A30

25. CJM 2003 (vol 55 pp. 1000)

Graczyk, P.; Sawyer, P.
Some Convexity Results for the Cartan Decomposition
In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$ where $a(g)$ is the abelian part in the Cartan decomposition of $g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$, $\mathbf{C}$ or $\mathbf{H}$. In particular, we show that $\mathcal{S}$ is convex. We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.

Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values
Categories:43A90, 53C35, 15A18
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