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26. CJM 2002 (vol 54 pp. 571)

Li, Chi-Kwong; Poon, Yiu-Tung
Diagonals and Partial Diagonals of Sum of Matrices
Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a certain group action such as \begin{enumerate}[(4)] \item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices $A$ by $(U,V)*A = UAV^t$, \item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m \times n$ real matrices $A$ by $(U,V)*A = UAV^t$, \item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$, \item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$. \end{enumerate} Denote by $$ \mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in \mathcal{O}(A_i), i = 1,\dots,k\} $$ the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}(A_1,\dots,A_k)$, {\it i.e.}, the set of vectors $(d_1,\dots,d_r)$ whose entries lie in the $(1,j_1),\dots,(r,j_r)$ positions of a matrix in $\mathcal{O}(A_1, \dots,A_k)$ for some distinct column indices $j_1,\dots,j_r$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of $A_1,\dots,A_k$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}(A_1,\dots,A_k)$ of complex Hermitian matrices under the action of unitary similarities are also discussed.

Keywords:orbit, group actions, unitary, orthogonal, Hermitian, (skew-)symmetric matrices, diagonal, singular values
Categories:15A42, 15A18

27. CJM 2001 (vol 53 pp. 758)

Goulden, I. P.; Jackson, D. M.; Latour, F. G.
Inequivalent Transitive Factorizations into Transpositions
The question of counting minimal factorizations of permutations into transpositions that act transitively on a set has been studied extensively in the geometrical setting of ramified coverings of the sphere and in the algebraic setting of symmetric functions. It is natural, however, from a combinatorial point of view to ask how such results are affected by counting up to equivalence of factorizations, where two factorizations are equivalent if they differ only by the interchange of adjacent factors that commute. We obtain an explicit and elegant result for the number of such factorizations of permutations with precisely two factors. The approach used is a combinatorial one that rests on two constructions. We believe that this approach, and the combinatorial primitives that have been developed for the ``cut and join'' analysis, will also assist with the general case.

Keywords:transitive, transposition, factorization, commutation, cut-and-join
Categories:05C38, 15A15, 05A15, 15A18

28. CJM 2001 (vol 53 pp. 470)

Bauschke, Heinz H.; Güler, Osman; Lewis, Adrian S.; Sendov, Hristo S.
Hyperbolic Polynomials and Convex Analysis
A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$ has all real roots for all vectors $x$. Motivated by partial differential equations, G{\aa}rding proved in 1951 that the largest such root is a convex function of $x$, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize G{\aa}rding's result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric function
Categories:90C25, 15A45, 52A41

29. CJM 2000 (vol 52 pp. 141)

Li, Chi-Kwong; Tam, Tin-Yau
Numerical Ranges Arising from Simple Lie Algebras
A unified formulation is given to various generalizations of the classical numerical range including the $c$-numerical range, congruence numerical range, $q$-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized.

Keywords:numerical range, convexity, inclusion relation
Categories:15A60, 17B20

30. CJM 2000 (vol 52 pp. 197)

Radjavi, Heydar
Sublinearity and Other Spectral Conditions on a Semigroup
Subadditivity, sublinearity, submultiplicativity, and other conditions are considered for spectra of pairs of operators on a Hilbert space. Sublinearity, for example, is a weakening of the well-known property~$L$ and means $\sigma(A+\lambda B) \subseteq \sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The effect of these conditions is examined on commutativity, reducibility, and triangularizability of multiplicative semigroups of operators. A sample result is that sublinearity of spectra implies simultaneous triangularizability for a semigroup of compact operators.

Categories:47A15, 47D03, 15A30, 20A20, 47A10, 47B10

31. CJM 1999 (vol 51 pp. 506)

Elduque, A.; Iltyakov, A. V.
On Polynomial Invariants of Exceptional Simple Algebraic Groups
We study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system (\cf\ \cite{Schw1}, \cite{Schw2}, \cite{Ilt}).

Categories:15A72, 17C20

32. CJM 1998 (vol 50 pp. 1323)

Morales, Jorge
L'invariant de Hasse-Witt de la forme de Killing
Nous montrons que l'invariant de Hasse-Witt de la forme de Killing d'une alg{\`e}bre de Lie semi-simple $L$ s'exprime {\`a} l'aide de l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de $L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines positives), et des invariants associ{\'e}s au groupe des sym{\'e}tries du diagramme de Dynkin de $L$.

Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66

33. CJM 1998 (vol 50 pp. 929)

Broer, Abraham
Decomposition varieties in semisimple Lie algebras
The notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements. We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, \eg, its normalization has rational singularities. The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class.

Categories:14L30, 14M17, 15A30, 17B45

34. CJM 1997 (vol 49 pp. 865)

Goulden, I. P.; Jackson, D. M.
Maps in locally orientable surfaces and integrals over real symmetric surfaces
The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surfaces. This complements the analogous result for monopoles in orientable surfaces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials.

Categories:05C30, 05A15, 05E05, 15A52

35. CJM 1997 (vol 49 pp. 840)

Rodman, Leiba
Non-Hermitian solutions of algebraic Riccati equation
Non-hermitian solutions of algebraic matrix Riccati equations (of the continuous and discrete types) are studied. Existence is proved of non-hermitian solutions with given upper bounds of the ranks of the skew-hermitian parts, under the sign controllability hypothesis.

Categories:15A99, 15A63, 93C60
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