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Allan Merino

Publications and source records attributed to Allan Merino.

11 recordsLinked to original sources
arXiv2026

On The Linearization of Alternative Means

Alternative means have recently attracted considerable attention in matrix analysis and operator theory. In this paper, we investigate the linearization problem for alternative means, namely the question of determining when a mean can be expressed as an affine combination of the matrices under consideration. We first prove a conjecture of Choi, Kim, and Lim for the Wasserstein mean. More precisely, we show that the Wasserstein mean $\text{A} \diamond \text{B}$ is linearizable if and only if $\text{A}\text{B} = \text{B}\text{A}$ and $\left|\text{Spec}(\text{A}^{-1}\text{B})\right| \leq 2$. We further establish a general rigidity theorem for a large class of alternative means. Specifically, we prove that an analogous characterization holds whenever the representing function $f$ is of the form $f(x) = \sqrt{h(x)}$, where $h$ is a non-affine operator monotone function. As consequences, we obtain linearization criteria for several families of alternative means, including logarithmic, harmonic, and power-type means.

math.FA↗
arXiv2026

Spectral Decomposition and Linearization of Kubo-Ando Means

In this paper, we study the structure of Kubo-Ando means on the cone of positive Hermitian matrices over the real numbers, complex numbers, and quaternions. Given a Kubo-Ando mean $\sigma$ with representing function $f$, we obtain an explicit decomposition of $\text{A} \sigma \text{B}$ in terms of the spectrum of $\text{A}^{-1}\text{B}$. More precisely, we show that $\text{A} \sigma \text{B}$ can be expressed as a finite linear combination of matrices of the form $\text{A}\left(\text{A}^{-1}\text{B}\right)^{k}$, with coefficients depending only on $f$ and the eigenvalues of $\text{A}^{-1}\text{B}$. We first investigate the linear case and characterize the pairs of matrices for which every Kubo-Ando mean admits an affine representation. We then focus on the cone $\mathscr{P}_{3}(\mathbb{D})$, where we derive explicit formulas for the decomposition coefficients in terms of spectral invariants. Finally, we show that the same techniques extend to a broad class of alternative means, yielding explicit decompositions in the commutative setting and extending recent results of Choi, Kim, and Lim.

math.FA↗
arXiv2026

Correspondence of Kubo-Ando Means over Real Division Algebras and Linearization of Means

In this paper, we establish a bijection between Kubo-Ando operator means defined on the cones $\mathscr{P}_{n}(\mathbb{H})$, $\mathscr{P}_{2n}(\mathbb{C})$, and $\mathscr{P}_{4n}(\mathbb{R})$. This correspondence is induced by the canonical embeddings relating quaternionic, complex, and real positive definite matrices. We investigate several structural and geometric properties preserved by these bijections, including compatibility with functional calculus, invariance under congruence transformations, and behavior with respect to natural metrics on these cones. As an application, we prove that every Kubo-Ando mean on $\mathscr{P}_{2}(\mathbb{D})$, where $\mathbb{D}\in\left\{\mathbb{R},\mathbb{C},\mathbb{H}\right\}$, admits an explicit affine expression in terms of the matrices involved. Using the embeddings above, we derive explicit formulas for operator means on special classes of real $4\times4$ positive definite matrices arising as images of the cones $\mathscr{P}_{2}(\mathbb{C})$ and $\mathscr{P}_{2}(\mathbb{H})$. In particular, we obtain trace-determinant formulas for the geometric mean in the real, complex, and quaternionic settings.

math.FA↗
arXiv2026

The Cone of J-Hermitian Matrices and a Geometric Mean

We study the cone $\mathscr{P}_{\text{J}}$ of positive J-Hermitian matrices associated with an indefinite signature matrix J = $\text{Id}_{p,q}$. We show that the J-exponential map is bijective and use it to analyze the algebraic and geometric structure of $\mathscr{P}_{\text{J}}$. Through a canonical identification with the cone of positive definite matrices, we endow $\mathscr{P}_{\text{J}}$ with a natural Riemannian structure. In this setting, we define a J-geometric mean as the midpoint of geodesics and prove that it is uniquely characterized as the solution of a Riccati-type equation.

math.DG↗
arXiv2025

Howe duality for the dual pair $\left(\text{SpO}(2n|1)\,, \mathfrak{osp}(2|2)\right)$

The goal of our work is to study the decomposition of the joint action of $\mathscr{G} = \text{SpO}(2n|1)$ and $\mathfrak{g}' = \mathfrak{osp}(2|2)$ on the supersymmetric algebra $\text{S} = \text{S}(\mathbb{C}^{2n|1} \otimes \mathbb{C}^{1|1})$. As proved by Merino and Salmasian, we have a one-to-one correspondence between irreducible representations of $\mathscr{G}$ and $\mathfrak{g}'$ appearing as subrepresentations of $\text{S}$. In this paper, we obtained an explicit description of the highest weights and joint highest weight vectors for the representations of $\mathscr{G}$ and $\mathfrak{g}'$ appearing in the duality.

math.RT↗
arXiv2022

Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup

In this paper, we obtain a full classification of reductive dual pairs in a, real or complex, Lie superalgebra $\mathfrak{spo}(E)$ and Lie supergroup $\textbf{SpO}(E)$. Moreover, by looking at the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(E)$ on the Weyl-Clifford algebra $\textbf{WC}(E)$, we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((G\,, \mathfrak{g})\,, (G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(E)$, the superalgebra $\textbf{WC}(E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in $\textbf{WC}(E)$ is generated by the Lie superalgebra $\mathfrak{g}'$. We obtain a full classification of reductive dual pairs in the (real or complex) Lie superalgebra $\mathfrak{spo}(\mathrm E)$ and the Lie supergroup $\textbf{SpO}(\mathrm E)$. Using this classification we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((\mathrm G\,, \mathfrak{g})\,, (\mathrm G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(\mathrm E)$, the superalgebra $\textbf{WC}(\mathrm E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in the Weyl-Clifford algebra $\textbf{WC}(\mathrm E)$, equipped with the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(\mathrm E)$, is generated by the Lie superalgebra $\mathfrak{g}'$. As an application, we prove that Howe duality holds for the dual pairs $({\textbf{SpO}}(2n|1)\,, {\textbf{OSp}}(2k|2l)) \subseteq {\textbf{SpO}}(\mathbb{C}^{2k|2l} \otimes \mathbb{C}^{2n|1})$.

math.RT↗
arXiv2019

Transfer of characters in the theta correspondence with one compact member

For an irreducible dual pair $(G, G') \in Sp(W)$ with one member compact and two representations $Π\leftrightarrow Π'$ appearing in the Howe duality, we give an expression of the character $Θ_{Π'}$ of $Π'$ via the character of $Π$. We make computations for the dual pair $(G = U(n, \mathbb{C}), G' = U(p, q, \mathbb{C}))$, which are explicit in low dimensions. For $(G = U(1, \mathbb{C}), G' = U(1, 1, \mathbb{C}))$, we verify directly a result of H. Hecht saying that the character has the same value on both Cartan subgroups of $G'$.

math.RT↗
arXiv2019

Dual pairs in the Pin-group and duality for the corresponding spinorial representation

In this paper, we give a complete picture of Howe correspondence for the setting ($O(E, b), Pin(E, b), Π$), where $O(E, b)$ is an orthogonal group (real or complex), $Pin(E, b)$ is the two-fold Pin-covering of $O(E, b)$, and $Π$ is the spinorial representation of $Pin(E, b)$. More precisely, for a dual pair ($G, G'$) in $O(E, b)$, we determine explicitly the nature of its preimages $(\tilde{G}, \tilde{G'})$ in $Pin(E, b)$, and prove that apart from some exceptions, $(\tilde{G}, \tilde{G'})$ is always a dual pair in $Pin(E, b)$; then we establish the Howe correspondence for $Π$ with respect to $(\tilde{G}, \tilde{G'})$.

math.RT↗
arXiv2019

Characters of some unitary highest weight representations via the theta correspondence

In this article, we consider a dual pair $(G, G')$ in the symplectic group $Sp(W)$ with $G$ compact and let $(\tilde{G}, \tilde{G}')$ be the preimages of $G$ and $G'$ in the metaplectic group $\widetilde{Sp(W)}$. For every irreducible representation $Π$ of $\tilde{G}$ appearing in Howe correspondence, we compute explicitly the restriction of the character $Θ_{Π'}$ of the associated representation $Π'$ of $\tilde{G}'$ on the set of regular points on the compact Cartan subgroup $\tilde{H}'$ of $\tilde{G}'$.

math.RT↗