Computational Invariant Theory

Isotropic invariants of a completely symmetric third-order tensor

“…Hence, the cardinal of a minimal integrity basis is generally greater than that of a functional basis. In mathematics, an irreducible functional basis is called a separating set 11 , but if their conciseness is appealing, no general algorithm currently exists to produce them. Let us now do a quick review on the state-of-the-art in invariant functions modeling in continuum mechanics.…”

Classifying semisimple orbits of $\theta $-groups

“…Being already available algorithms to compute the fundamental invariants of a complex reflection groups given its generators (see [5], [21]), my work was reduced, essentially, to the above points.…”

Communication: Fitting potential energy surfaces with fundamental invariant neural network

“…It is worthwhile to point out that the mathematical background of fundamental invariants was well established decades ago. 20 Recently, Opalka and Domcke employed FI to generate invariant polynomials in their linear invariant polynomial fitting of multi-sheeted PESs for a XY 4 molecule. 21 For a molecular system A i B j · · · X p , internuclear distances (r 1 ,r 2 , .…”

“…R G is finitely generated and has a minimal set of homogeneous generators called fundamental invariants. 20 The fundamental invariants can be calculated with King's algorithm 22 implemented in the computer algebra system called Singular. 23 Let R G denote the invariant ring of a finite group G. For any f ∈ R G , there exists a set of invariants P = {p i ∈ R G , 1 i n} (called as primary invariants), a set of invariants S = {θ j ∈ R G , 1 j t} (called as secondary invariants), and a unique set of polynomials…”

“…Equation (1) is the expression of Hironaka decomposition, 20,24 which states that primary and secondary invariants together can be used as the generators of R G . Here R[p 1 , .…”