Current research on algebraic combinatorics

1986 ◽  
Vol 2 (1) ◽  
pp. 287-308 ◽  
Author(s):  
Eiichi Bannai ◽  
Tatsuro Ito
Keyword(s):  
Algebraic Combinatorics

Dodgson condensation, alternating signs and square ice

2006 ◽  
Vol 364 (1849) ◽  
pp. 3183-3198 ◽  
Author(s):  
Andrew N.W Hone
Keyword(s):  
Number Theory ◽  
Computer Algebra ◽  
Statistical Physics ◽  
Mathematical Physics ◽  
Algebraic Combinatorics ◽  
General Scientific ◽  
Dodgson Condensation

Starting with the numbers 1,2,7,42,429,7436, what is the next term in the sequence? This question arose in the area of mathematics called algebraic combinatorics, which deals with the precise counting of sets of objects, but it goes back to Lewis Carroll's work on determinants. The resolution of the problem was only achieved at the end of the last century, and with two completely different approaches: the first involved extensive verification by computer algebra and a huge posse of referees, while the second relied on an unexpected connection with the theory of ‘square ice’ in statistical physics. This paper, aimed at a general scientific audience, explains the background to this problem and how subsequent developments are leading to a fruitful interplay between algebraic combinatorics, mathematical physics and number theory.

Algebraic combinatorics and QFT

2021 ◽  
pp. 76-94
Author(s):  
Adrian Tanasa
Keyword(s):  
Hopf Algebra ◽  
Hochschild Cohomology ◽  
Algebraic Combinatorics ◽  
Fourth Section ◽  
Analytic Combinatorics ◽  
Multi Scale ◽  
Starting Point ◽  
Feynman Graphs ◽  
Perturbative Renormalization

We have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).

Quantum gravity, group field theory (GFT), and combinatorics

2021 ◽  
pp. 121-165
Author(s):  
Adrian Tanasa
Keyword(s):  
Field Theory ◽  
Quantum Theory ◽  
Quantum Gravity ◽  
Specific Model ◽  
Algebraic Combinatorics ◽  
Saddle Point Method ◽  
Feynman Integrals ◽  
Causal Dynamical Triangulations ◽  
Group Field Theory ◽  
Definition Of

This chapter is the first chapter of the book dedicated to the study of the combinatorics of various quantum gravity approaches. After a brief introductory section to quantum gravity, we shortly mention the main candidates for a quantum theory of gravity: string theory, loop quantum gravity, and group field theory (GFT), causal dynamical triangulations, matrix models. The next sections introduce some GFT models such as the Boulatov model, the colourable and the multi-orientable model. The saddle point method for some specific GFT Feynman integrals is presented in the fifth section. Finally, some algebraic combinatorics results are presented: definition of an appropriate Conne–Kreimer Hopf algebra describing the combinatorics of the renormalization of a certain tensor GFT model (the so-called Ben Geloun–Rivasseau model) and the use of its Hochschild cohomology for the study of the combinatorial Dyson–Schwinger equation of this specific model.

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