Hall polynomial

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The Hall polynomials in mathematics were developed by Philip Hall in the 1950s in the study of group representations. These polynomials are the structure constants of a certain associative algebra, called the Hall algebra, which plays an important role in the theory of Kashiwara-Lusztig's canonical bases in quantum groups.

[edit] Construction

A finite abelian p-group M is a direct sum of cyclic p-power components C_{p^\lambda_i}, where \lambda=(\lambda_1,\lambda_2,\ldots) is a partition of n called the type of M. Let g^\lambda_{\mu,\nu}(p) be the number of subgroups N of M such that N has type ν and the quotient M/N has type μ. Hall proved that the functions g are polynomial functions of p with integer coefficients. Thus we may replace p with an indeterminate q, which results in the Hall polynomials

g^\lambda_{\mu,\nu}(q)\in\mathbb{Z}[q].

Hall next constructs an associative ring H over \mathbb{Z}[q], now called the Hall algebra. This ring has a basis consisting of the symbols uλ and the structure constants of the multiplication in this basis are given by the Hall polynomials:

u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu}(q) u_\lambda.

It turns out that H is a commutative ring, freely generated by the elements u_{\mathbf1^n} corresponding to the elementary p-groups. The linear map from H to the algebra of symmetric functions defined on the generators by the formula

u_{\mathbf 1^n} \mapsto q^{-n(n-1)}e_n

(where en is the nth elementary symmetric function) uniquely extends to a ring homomorphism and the images of the basis elements uλ may be interpreted via the Hall-Littlewood symmetric functions. Specializing q to 1, these symmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials.

[edit] References