Symmetry of Dressed Photon

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n≥r≥s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with n=4,r=2,s=1.

A NOTE ON RATIONAL HOMOLOGICAL STABILITY OF AUTOMORPHISMS OF MANIFOLDS

By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp (S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $*\le (g-6)/2$. In this note, we explain how this range can be improved to $*\le g-2$ using cohomological vanishing results due to Borel and the classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.

Poincare series for the algebras of joint invariants and covariants of $n$ quadratic forms

We consider one of the fundamental problems of classical invariant theory - the research of Poincare series for an algebra of invariants of Lie group $SL_2$. The first two terms of the Laurent series expansion of Poincare series at the point $z = 1$ give us important information about the structure of the algebra $\mathcal{I}_{d}.$ It was derived by Hilbert for the algebra ${\mathcal{I}_{d}=\mathbb{C}[V_d]^{\,SL_2}}$ of invariants for binary $d-$form (by $V_d$ denote the vector space over $\mathbb{C}$ consisting of all binary forms homogeneous of degree $d$). Springer got this result, using explicit formula for the Poincare series of this algebra. We consider this problem for the algebra of joint invariants $\mathcal{I}_{2n}=\mathbb{C}[\underbrace{V_2 \oplus V_2 \oplus \cdots \oplus V_2}_{\text{n times}}]^{SL_2}$ and the algebra of joint covariants $\mathcal{C}_{2n}=\mathbb{C}[\underbrace{V_2 {\oplus} V_2 {\oplus} \cdots {\oplus} V_2}_{\text{n times}}{\oplus}\mathbb{C}^2 ]^{SL_2}$ of $n$ quadratic forms. We express the Poincare series $\mathcal{P}(\mathcal{C}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{C}_{2n})_{j}\, z^j$ and $\mathcal{P}(\mathcal{I}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{I}_{2n})_{j}\, z^j$ of these algebras in terms of Narayana polynomials. Also, for these algebras we calculate the degrees and asymptotic behavious of the degrees, using their Poincare series.