Hall algebra

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 $C_{p_{i}^{\lambda }},$ where $\lambda =(\lambda _{1},\lambda _{2},\ldots )$ is a partition of $n$ called the type of M. Let $g_{\mu ,\nu }^{\lambda }(p)$ be the number of subgroups N of M such that N has type $\nu$ 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

$g_{\mu ,\nu }^{\lambda }(q)\in \mathbb {Z} [q].$

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 $u_{\lambda }$ and the structure constants of the multiplication in this basis are given by the Hall polynomials:

$u_{\mu }u_{\nu }=\sum _{\lambda }g_{\mu ,\nu }^{\lambda }(q)u_{\lambda }.$

It turns out that H is a commutative ring, freely generated by the elements $u_{\mathbf {1} ^{n}}$ 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 nth 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.