mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Macdonald identities are some infinite product identities associated to

affine root system In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple ''p''-adic algebraic groups, and correspond to f ...

s, introduced by . They include as special cases the

Jacobi triple product identity In mathematics, the Jacobi triple product is the mathematical identity: :\prod_^\infty \left( 1 - x^\right) \left( 1 + x^ y^2\right) \left( 1 +\frac\right) = \sum_^\infty x^ y^, for complex numbers ''x'' and ''y'', with , ''x'', < 1 and ''y' ...

, Watson's quintuple product identity, several identities found by , and a 10-fold product identity found by . and pointed out that the Macdonald identities are the analogs of the

Weyl denominator formula In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...

for

affine Kac–Moody algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...

s and superalgebras.

References

* * * * * *{{Citation , last1=Winquist , first1=Lasse , title=An elementary proof of p(11m+6) ≡ 0 mod 11 , mr=0236136 , year=1969 , journal=Journal of Combinatorial Theory , volume=6 , pages=56–59 , doi=10.1016/s0021-9800(69)80105-5, doi-access=free Lie algebras Mathematical identities Infinite products