Irreducible function bases of isotropic invariants of a third order three-dimensional symmetric and traceless tensor

Abstract

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order threedimensional symmetric and traceless tensor has four invariants with degrees two, four, six and ten respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no polynomial syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.Key words. minimal integrity basis, irreducible function basis, symmetric and traceless tensor, syzygy. Nomenclature D a third order three-dimensional symmetric and traceless tensor with components D ijk T(m, n) the space of real tensors of order m and dimension n S(m, n) the subspace of symmetric tensors St(m, n) the subspace of symmetric and traceless tensors O(n) the orthogonal group of dimension n SO(n) the special orthogonal group of dimension n Gl(n, R) the general linear group of real matrices m n = m! n!(m−n)! the binomial coefficient for m ≥ n ≥ 0The next theorem claims that there is no syzygy relation among four invariants J 2 , J 4 , J 6 and J 10 , where {J 2 , J 4 , J 6 , J 10 } be an arbitrary minimal integrity basis of D.Theorem 4.1. Let {J 2 , J 4 , J 6 , J 10 } be an arbitrary minimal integrity basis of a third order three-dimensional symmetric and traceless tensor D. Then there is no syzygy relation among four invariants J 2 , J 4 , J 6 and J 10 .

An irreducible function basis of isotropic invariants of a third order three-dimensional symmetric tensor

“…First, this is the first result on irreducible function bases of a third order three-dimensional symmetric tensor. Second, there are still at least three syzygy relations among these eleven invariants, see (3)(4)(5). This shows that an irreducible function basis consisting of polynomial invariants may not be algebraically minimal in the sense that the basis consists of polynomial invariants and there is no algebraic relations in these invariants [10].…”

“…Note that Boehler, Kirillov and Onat [2] had already given a minimal integrity basis for a fourth order three-dimensional symmetric and traceless tensor in 1994. But the minimal integrity basis given by Smith and Bao [11] for the same tensor is slightly different [3]. In 2014, an integrity basis with thirteen isotropic invariants of a (completely) symmetric third order three-dimensional tensor was presented by Olive and Auffray [6].…”

Isotropic polynomial invariants of Hall tensor

“…Next, we need to verify the polynomial irreducibility of this integrity basis. A natural observation is that these isotropic invariants are homogenous polynomials of the 9 independent components in the Hall tensor K. A similar approach as the method proposed by Chen et al [12] is employed in this part.…”

“…Furthermore, Boehler et al [2] investigate polynomial invariants for the elasticity tensor in 1993. Some very recent works are devoted to minimal integrity bases and irreducible function bases for third order and fourth order tensors [12,13] .…”

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

“…Liu, Ding, Qi and Zou [11] gave a minimal integrity basis and irreducible functional basis of isotropic invariants of the Hall tensors. Chen, Hu, Qi and Zou [4] showed that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is indeed an irreducible functional basis of that tensor. Chen, Liu, Qi, Zheng and Zou [5] presented an eleven invariant irreducible functional basis for a third order three-dimensional symmetric tensor.…”