arXiv subjects

G. Valente

Publications and source records attributed to G. Valente.

23 records · Page 2Linked to original sources

arXiv2003

Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata

Functions which are equivariant or invariant under the transformations of a compact linear group $G$ acting in an euclidean space $\real^n$, can profitably be studied as functions defined in the orbit space of the group. The orbit space is the union of a finite set of strata, which are semialgebraic manifolds formed by the $G$-orbits with the same orbit-type. In this paper we provide a simple recipe to obtain rational parametrizations of the strata. Our results can be easily exploited, in many physical contexts where the study of equivariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau theory), in equivariant bifurcation theory, in crystal field theory and in most areas where use is made of symmetry adapted functions. A physically significant example of utilization of the recipe is given, related to spontaneous polarization in chiral biaxial liquid crystals, where the advantages with respect to previous heuristic approaches are shown.

math-ph↗

arXiv2001

Possible ground states of D-wave condensates in isotropic space

A complete and rigorous determination of the possible ground states for D-wave pairing Bose condensates is presented. Using an orbit space approach to the problem, we find 15 allowed phases (besides the unbroken one), with different symmetries, that we thoroughly determine, specifying the group-subgroup relations between bordering phases.

cond-mat.supr-con↗

arXiv2001

Geometric invariant theory approach to the determination of ground states of D-wave condensates in isotropic space

A complete and rigorous determination of the possible ground states for D-wave pairing Bose condensates is presented, using a geometrical invariant theory approach to the problem. The order parameter is argued to be a vector, transforming according to a ten dimensional real representation of the group $G=${\bf O}$_3\otimes${\bf U}$_1\times <{\cal T}>$. We determine the equalities and inequalities defining the orbit space of this linear group and its symmetry strata, which are in a one-to-one correspondence with the possible distinct phases of the system. We find 15 allowed phases (besides the unbroken one), with different symmetries, that we thoroughly determine. The group-subgroup relations between bordering phases are pointed out. The perturbative sixth degree corrections to the minimum of a fourth degree polynomial $G$-invariant free energy, calculated by Mermin, are also determined.

cond-mat.supr-con↗

arXiv2000

Determination of the orbit spaces of non-coregular compact linear groups with one relation among the basic polynomial invariants in the $\hat P$-matrix approach

Invariant functions under the transformations of a compact linear group $G$ acting in $\real^n$ can be expressed in terms of functions defined in the orbit space of $G$. We develop a method to determine the isotropy classes of the orbit spaces of all the real linear groups whose integrity bases (IB) satisfy only one independent relation. The method is tested for IB's formed by 3 (independent) basic invariants. The result is obtained through a metric matrix $\hat P(p)$, defined only from the scalar products between the gradients of a minimal IB. We determine the matrices $\wP(p)$ from a universal differential equation, which satisfy new convenient additional conditions, which fit for the non-coregular case. Our results may be relevant in physical contexts where the study of covariant or invariant functions is important, like in the determination of patterns of spontaneous symmetry breaking in quantum field theory, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and so on.

hep-th↗

arXiv1995

Orbit spaces of reflection groups with 2, 3, and 4 basic polynomial invariants

Covariant or invariant functions under a compact linear group can be expressed in terms of functions defined in the orbit space of the group. The semialgebraic relations defining the orbit spaces of all finite coregular real linear groups with at most 4 basic invariants are determined. For each group $G$ acting in $\real^n$, the results are obtained through the computation of a metric matrix $\widehat P(p)$, which is defined only in terms of the scalar products between the gradients of a set of basic polynomial invariants $p_1(x),\dots p_q(x),\x\in\real^n$ of $G$; the semi-positivity conditions $\widehat P(p)\ge 0$ are known to determine all the equalities and inequalities defining the orbit space $\real^n/G$ of $G$ as a semi-algebraic variety in the space $\real^q$ spanned by the variables $p_1,\dots ,p_q$. In a recent paper, the $\widehat P$-matrices, for $q\le 4$, have been determined in an alternative way, as solutions of a universal differential equation;the present paper yields a partial, but significant, check on the correctness and completeness of these solutions. Our results can be widely exploited,e.g. in the determination of patterns of spontaneous symmetry breaking, in the analysis of structural phase transitions (Landau's theory),in covariant bifurcation theory,in crystal field theory and in solid state theory where symmetry adapted functions are used.

hep-th↗