𝔖 Bobbio Scriptorium
✦   LIBER   ✦

A Stable Range for Dimensions of Homogeneous O(n)-Invariant Polynomials on the n × n Matrices

✍ Scribed by Jeb F Willenbring

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
153 KB
Volume
242
Category
Article
ISSN
0021-8693

No coin nor oath required. For personal study only.

✦ Synopsis

The complex orthogonal group O n acts on the n × n matrices, M n , by restricting the adjoint action of GL n

. This action provides us with an action on the ring of complex valued polynomial functions on the n × n matrices, M n . The polynomials of degree d, denoted d M n , form a finite dimensional representation of O n and provide a graded module structure on M n as well as the subring of invariant polynomials,

is equal to the coefficient of q d in ∞ k=1 1/ 1 -q k c k , where c k is the number of k vertex cyclic graphs with directed edges counted up to dihedral symmetry. The above formula provides a combinatorial interpretation of an initial segment of the Hilbert series for this ring.

📜 SIMILAR VOLUMES

A Note on the Riemann Problem for Genera
A Note on the Riemann Problem for General n × n Conservation Laws
✍ Fabio Ancona; Andrea Marson 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 119 KB

We analyze the structure of the general solution of the Riemann problem for a strictly hyperbolic system of conservation laws whose characteristic fields are neither genuinely non-linear nor linearly degenerate in the sense of Lax.