scholarly journals Generalized Littlewood–Richardson coefficients for branching rules of GL(n) and extremal weight crystals

2020 ◽  
Vol 3 (6) ◽  
pp. 1365-1400
Author(s):  
Brett Collins
Keyword(s):  
Branching Rules

Lie Groups and the Jahn–Teller Effect

1974 ◽  
Vol 52 (11) ◽  
pp. 999-1044 ◽  
Author(s):  
B. R. Judd
Keyword(s):  
Lie Groups ◽  
Nuclear Physics ◽  
Angular Frequency ◽  
Teller Effect ◽  
Octahedral Complex ◽  
Noncompact Group ◽  
Five Dimensions ◽  
Branching Rules ◽  
Jahn Teller ◽  
Jahn Teller Effect

After an introduction to the classic theory of the Jahn–Teller effect for octahedral complexes, an account is given of Lie groups and their relevance to the F+ center in CaO. The coincidence of the three-fold and two-fold vibrational modes (both of angular frequency ω) leads to a study of U5 and R5, the unitary and rotation groups in five dimensions. The language of second quantization is used to describe the weight spaces and branching rules. Pairs of annihilation and creation operators for phonons are coupled to zero angular momentum and used as the generators of the noncompact group O(2, 1). This facilitates the evaluation of matrix elements of V, the interaction that couples the oscillations of the octahedral complex to the electron in its interior. Glauber states are used near the strong Jahn–Teller limit, corresponding to [Formula: see text]. The possible extension of the analysis to incorporate the breathing mode is outlined. Correspondences with problems in nuclear physics are mentioned.

scholarly journals Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic

2017 ◽  
Vol 12 (2) ◽  
Author(s):  
Marilynn Johnson
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Variable Domain ◽  
Free Logic ◽  
Intuitionist Logic ◽  
Branching Rule ◽  
Branching Rules ◽  
Graham Priest

In An Introduction to Non-Classical Logic: From If to Is Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.

Branching rules for representations of classical Lie groups

1990 ◽  
Vol 84 (1) ◽  
pp. 675-682
Author(s):  
V. D. Lyakhovskii ◽  
I. A. Filanovskii
Keyword(s):  
Lie Groups ◽  
Branching Rules

scholarly journals Stable branching rules for classical symmetric pairs

2004 ◽  
Vol 357 (4) ◽  
pp. 1601-1626 ◽  
Author(s):  
Roger Howe ◽  
Eng-Chye Tan ◽  
Jeb F. Willenbring
Keyword(s):  
Branching Rules

Hopf Algebras from Branching Rules

1983 ◽  
Vol 35 (1) ◽  
pp. 177-192 ◽  
Author(s):  
P. Hoffman
Keyword(s):  
Abelian Group ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Homogeneous Element ◽  
Algebra Structure ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Dimension Zero ◽  
Branching Rules

Below we work out the algebra structure of some Hopf algebras which arise concretely in restricting representations of the symmetric group to certain subgroups. The basic idea generalizes that used by Adams [1] for H*(BSU). The question arose in discussions with H. K. Farahat. I would like to thank him for his interest in the work and to acknowledge the usefulness of several stimulating conversations with him.1. Review and statement of results. A homogeneous element of a graded abelian group will have its gradation referred to as its dimension. In all such groups below there will be no non-zero elements with negative or odd dimension. A graded algebra (resp. coalgebra) will be associative (resp. coassociative), strictly commutative (resp. co-commutative) and in dimension zero will be isomorphic to the ground ring F, providing the unit (resp. counit). We shall deal amost entirely with F = Z or F = Z/p for a prime p; the cases F = 0 or a localization of Z will occur briefly. In every case, the component in each dimension will be a finitely generated free F-module, so dualization works simply.

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