Branching rules, canonical bases and decomposition matrices

1999 ◽  
pp. 95-135
Author(s):  
Andrew Mathas
Keyword(s):  
Canonical Bases ◽  
Branching Rules

scholarly journals Combinatorics of canonical bases revisited: type A

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Volker Genz ◽  
Gleb Koshevoy ◽  
Bea Schumann
Keyword(s):  
Type A ◽  
Canonical Bases

scholarly journals Laurent Positivity of Quantized Canonical Bases for Quantum Cluster Varieties from Surfaces

2019 ◽  
Vol 373 (2) ◽  
pp. 655-705 ◽  
Author(s):  
So Young Cho ◽  
Hyuna Kim ◽  
Hyun Kyu Kim ◽  
Doeun Oh
Keyword(s):  
Canonical Bases ◽  
Quantum Cluster ◽  
Cluster Varieties

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 Factorization of the canonical bases for higher-level Fock spaces

2011 ◽  
Vol 55 (1) ◽  
pp. 23-51 ◽  
Author(s):  
Susumu Ariki ◽  
Nicolas Jacon ◽  
Cédric Lecouvey
Keyword(s):  
Fock Space ◽  
Fock Spaces ◽  
Transition Matrices ◽  
Canonical Bases ◽  
Highest Weight ◽  
Highest Weight Modules ◽  
Graphic Module ◽  
Module Structures ◽  
Weight Modules

AbstractThe level l Fock space admits canonical bases $\mathcal{G}_{e}$ and $\smash{\mathcal{G}_{\infty}}$. They correspond to $\smash{\mathcal{U}_{v}(\widehat{\mathfrak{sl}}_{e})}$ and $\mathcal{U}_{v}({\mathfrak{sl}}_{\infty})$-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in ℕ[v]. Restriction to the highest-weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki–Koike algebras.

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.

Sign in / Sign up

Export Citation Format

Share Document