“MacMahon Master theorem”的意思、由来-开放百科全书

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MacMahon Master theorem

释义

Background
Precise statement
Derivation of Dixon's identity
See also
References

In mathematics, the MacMahon Master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis (1916). It is often used to derive binomial identities, most notably Dixon's identity.

Background

In the monograph, MacMahon found so many applications of his result, he called it "a master theorem in the Theory of Permutations." He explained the title as follows: "a Master Theorem from the masterly and rapid fashion in which it deals with various questions otherwise troublesome to solve."

The result was re-derived (with attribution) a number of times, most notably by I. J. Good who derived it from his multilinear generalization of the Lagrange inversion theorem. MMT was also popularized by Carlitz who found an exponential power series version. In 1962, Good found a short proof of Dixon's identity from MMT. In 1969, Cartier and Foata found a new proof of MMT by combining algebraic and bijective ideas (built on Foata's thesis) and further applications to combinatorics on words, introducing the concept of traces. Since then, MMT has become a standard tool in enumerative combinatorics.

Although various q-Dixon identities have been known for decades, except for a Krattenthaler–Schlosser extension (1999), the proper q-analog of MMT remained elusive. After Garoufalidis–Lê–Zeilberger's quantum extension (2006), a number of noncommutative extensions were developed by Foata–Han, Konvalinka–Pak, and Etingof–Pak. Further connections to Koszul algebra and quasideterminants were also found by Hai–Lorentz, Hai–Kriegk–Lorenz, Konvalinka–Pak, and others.

Finally, according to J. D. Louck, theoretical physicist Julian Schwinger re-discovered the MMT in the context of his generating function approach to the angular momentum theory of many-particle systems. Louck writes:

{{quote|It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary build-up of such systems from more elementary constituents.^[1]}}

Precise statement

Let be a complex matrix, and let be formal variables. Consider a coefficient

(Here the notation means "the coefficient of monomial in ".) Let be another set of formal variables, and let be a diagonal matrix. Then

where the sum runs over all nonnegative integer vectors ,

and denotes the identity matrix of size .

Derivation of Dixon's identity

Consider a matrix

Compute the coefficients G(2n, 2n, 2n) directly from the definition:

where the last equality follows from the fact that on the right-hand side we have the product of the following coefficients:

which are computed from the binomial theorem. On the other hand, we can compute the determinant explicitly:

Therefore, by the MMT, we have a new formula for the same coefficients:

where the last equality follows from the fact that we need to use an equal number of times all three terms in the power. Now equating the two formulas for coefficients G(2n, 2n, 2n) we obtain an equivalent version of Dixon's identity:

References

1. ^{{cite book|last1=Louck|first1=James D.|title=Unitary symmetry and combinatorics|date=2008|publisher=World Scientific|location=Singapore|isbn=978-981-281-472-2|pages=viii}}

P.A. MacMahon, Combinatory analysis, vols 1 and 2, Cambridge University Press, 1915–16.
{{cite journal | zbl=0108.25104 | authorlink=I. J. Good | first=I.J. | last=Good | title=A short proof of MacMahon's ‘Master Theorem’ | journal=Proc. Cambridge Philos. Soc. | volume=58 | year=1962 | page=160 }}
{{cite journal | zbl=0108.25105 | authorlink=I. J. Good | first=I.J. | last=Good | title=Proofs of some `binomial' identities by means of MacMahon's ‘Master Theorem’ | journal=Proc. Cambridge Philos. Soc. | volume=58 | year=1962 | pages=161–162 }}
P. Cartier and D. Foata, Problèmes combinatoires de commutation et réarrangements, Lecture Notes in Mathematics, no. 85, Springer, Berlin, 1969.
L. Carlitz, An Application of MacMahon's Master Theorem, SIAM Journal on Applied Mathematics 26 (1974), 431–436.
I.P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley, New York, 1983.
C. Krattenthaler and M. Schlosser, A new multidimensional matrix inverse with applications to multiple q-series, Discrete Math. 204 (1999), 249–279.
S. Garoufalidis, T. T. Q. Lê and D. Zeilberger, The Quantum MacMahon Master Theorem, Proc. Natl. Acad. of Sci. 103 (2006), no. 38, 13928–13931 ([https://arxiv.org/abs/math/0303319 eprint]).
M. Konvalinka and I. Pak, Non-commutative extensions of the MacMahon Master Theorem, Adv. Math. 216 (2007), no. 1. ([https://arxiv.org/abs/math/0607737 eprint]).
D. Foata and G.-N. Han, A new proof of the Garoufalidis-Lê-Zeilberger Quantum MacMahon Master Theorem, J. Algebra 307 (2007), no. 1, 424–431 ([https://arxiv.org/abs/math/0603464 eprint]).
D. Foata and G.-N. Han, Specializations and extensions of the quantum MacMahon Master Theorem, Linear Algebra Appl 423 (2007), no. 2–3, 445–455 ([https://arxiv.org/abs/math.CO/0603466 eprint]).
P.H. Hai and M. Lorenz, Koszul algebras and the quantum MacMahon master theorem, Bull. Lond. Math. Soc. 39 (2007), no. 4, 667–676. ([https://arxiv.org/abs/math/0603169 eprint]).
P. Etingof and I. Pak, An algebraic extension of the MacMahon master theorem, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2279–2288 ([https://arxiv.org/abs/math/0608005 eprint]).
P.H. Hai, B. Kriegk and M. Lorenz, N-homogeneous superalgebras, J. Noncommut. Geom. 2 (2008) 1–51 ([https://arxiv.org/abs/0704.1888 eprint]).
J.D. Louck, Unitary symmetry and combinatorics, World Sci., Hackensack, NJ, 2008.

5 : Enumerative combinatorics|Factorial and binomial topics|Articles containing proofs|Theorems in combinatorics|Theorems in linear algebra

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Background

Precise statement

Derivation of Dixon's identity

See also

References