scholarly journals Symmetry of Dressed Photon

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1283
Author(s):  
Hiroyuki Ochiai
Keyword(s):  
Theoretical Model ◽  
Homogeneous Space ◽  
Lie Group ◽  
Invariant Theory ◽  
Grassmann Manifold ◽  
Vector Spaces ◽  
Classical Invariant Theory ◽  
Lie Group Actions ◽  
Prehomogeneous Vector Spaces ◽  
Classical Invariant

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.

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

2017 ◽  
Vol 9 (1) ◽  
pp. 57-62 ◽  
Author(s):  
N.B. Ilash
Keyword(s):  
Lie Group ◽  
Invariant Theory ◽  
Quadratic Forms ◽  
Laurent Series ◽  
Poincaré Series ◽  
Binary Forms ◽  
Poincare Series ◽  
Classical Invariant Theory ◽  
Joint Invariants ◽  
Classical Invariant

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.

Relationship to Classical Invariant Theory

2015 ◽  
pp. 129-142
Author(s):  
V. Lakshmibai ◽  
Justin Brown
Keyword(s):  
Invariant Theory ◽  
Classical Invariant Theory ◽  
Classical Invariant

scholarly journals ON THE VOLUME CONJECTURE FOR CLASSICAL SPIN NETWORKS

2012 ◽  
Vol 21 (03) ◽  
pp. 1250022 ◽  
Author(s):  
ABDELMALEK ABDESSELAM
Keyword(s):  
Invariant Theory ◽  
Classical Literature ◽  
Spin Networks ◽  
Classical Invariant Theory ◽  
Classical Spin ◽  
Volume Conjecture ◽  
Symbolic Method ◽  
Exact Determination ◽  
Classical Invariant

We prove an upper bound for the evaluation of all classical SU2 spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.

Classical Invariant Theory

2009 ◽  
pp. 225-300 ◽  
Author(s):  
Roe Goodman ◽  
Nolan R. Wallach
Keyword(s):  
Invariant Theory ◽  
Classical Invariant Theory ◽  
Classical Invariant

scholarly journals A NOTE ON RATIONAL HOMOLOGICAL STABILITY OF AUTOMORPHISMS OF MANIFOLDS

2020 ◽  
Vol 71 (3) ◽  
pp. 1069-1079
Author(s):  
Manuel Krannich
Keyword(s):  
Invariant Theory ◽  
Characteristic Classes ◽  
Classical Invariant Theory ◽  
Homological Stability ◽  
Classical Invariant

Abstract 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.

scholarly journals Symmetric Determinants and the Cayley and Capelli Operator

1948 ◽  
Vol 8 (2) ◽  
pp. 76-86 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Differential Operators ◽  
Matrix Operator ◽  
Linear Combinations ◽  
Classical Invariant Theory ◽  
Independent Variables ◽  
The Matrix ◽  
Classical Invariant ◽  
Initial Discovery ◽  
Special Case

The result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.

Hook length products and Cayley operators of classical invariant theory

1981 ◽  
Vol 9 (4) ◽  
pp. 289-297 ◽  
Author(s):  
E.R. Canfield ◽  
S.G. Williamson
Keyword(s):  
Invariant Theory ◽  
Hook Length ◽  
Classical Invariant Theory ◽  
Classical Invariant
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