Hall polynomial

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.

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 $g^lambda\_\{mu,\; u\}(p)$ be the number of subgroups "N" of "M" such that "N" has type $u$ and the quotient "M/N" has type $mu$. 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 : $g^lambda\_\{mu,\; u\}(q)inmathbb\{Z\}\; [q]\; .$

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 $u\_lambda$ and the structure constants of the multiplication in this basis are given by the Hall polynomials:

:$u\_mu\; u\_\; u\; =\; sum\_lambda\; g^lambda\_\{mu,\; u\}(q)\; u\_lambda.$

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 "e"_n is the "n"th elementary symmetric function) uniquely extends to a ring homomorphism and the images of the basis elements $u\_lambda$ 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.

References

* Ian G. Macdonald, "Symmetric functions and Hall polynomials", (Oxford University Press, 1979) ISBN 0-19-853530-9
* Claus Michael Ringel, "Hall algebras and quantum groups." Invent. Math. 101 (1990), no. 3, 583–591.
* George Lusztig, "Quivers, perverse sheaves, and quantized enveloping algebras." J. Amer. Math. Soc. 4 (1991), no. 2, 365–421.