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.

scholarly journals ON THE LINEAR COMBINANTS OF A BINARY PENCIL

2009 ◽  
Vol 51 (3) ◽  
pp. 481-498 ◽  
Author(s):  
ABDELMALEK ABDESSELAM ◽  
JAYDEEP CHIPALKATTI
Keyword(s):  
Angular Momentum ◽  
Quantum Theory ◽  
Invariant Theory ◽  
Binary Forms ◽  
Classical Invariant Theory ◽  
Symbolic Method ◽  
Classical Invariant ◽  
Explicit Formulae

AbstractLet A, B denote binary forms of order d, and let 2r−1 = (A, B)2r−1 be the sequence of their linear combinants for $1 \le r \le \lfloor\frac{d+1}{2}\rfloor$. It is known that 1, 3 together determine the pencil {A + λ B}λ∈P1 and hence indirectly the higher combinants 2r−1. In this paper we exhibit explicit formulae for all r ≥ 3, which allow us to recover 2r−1 from the knowledge of 1 and 3. The calculations make use of the symbolic method in classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the plethysm representation ∧2Sd for the group SL2. We give an example for the group SL3 to show that such a result may hold for other categories of representations.

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.

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