an:06588977
Zbl 1338.05272
Babaei, Esmaeil; Zamani, Yousef; Shahryari, Mohammad
Symmetry classes of polynomials
EN
Commun. Algebra 44, No. 4, 1514-1530 (2016).
00355669
2016
j
05E05 15A69 20C30
alternating group; irreducible characters; orthogonal basis; symmetry class of polynomials; symmetry class of tensors; symmetric group
Summary: Let \(G\) be a subgroup of \(S_m\) and suppose \(\chi\) is an irreducible complex character of \(G\). Let \(H_d(G,\chi)\) be the symmetry class of polynomials of degree \(d\) with respect to \(G\) and \(\chi\). Let \(V\) be an \((d+1)\)-dimensional inner product space over \(\mathbb{C}\) and \(V_\chi(G)\) be the symmetry class of tensors associated with \(G\) and \(\chi\). A monomorphism \(H_d(G,\chi)\to V_\chi(G)\) is given and it is used to obtain necessary and sufficient conditions for nonvanishing \(H_d(G,\chi)\). The nonexistence of o-basis of \(H_d(S_m,\chi^\pi)\) for a certain class of irreducible characters of \(S_m\) is concluded. The dimensions of symmetry classes of polynomials with respect to the irreducible characters of \(S_m\) and \(A_m\) are computed.