Hall algebra

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 \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^\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 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^\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)/2}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

• {{citation|first=Philip|last=Hall|authorlink=Philip Hall| year=1959|chapter=The algebra of partitions|title=Proceedings of the 4th Canadian mathematical congress, Banff|pages=147–159}}
• George Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, Journal of the American Mathematical Society 4 (1991), no. 2, 365–421.
• {{citation|contribution=Lectures on Hall algebras|last=Schiffmann|first=Olivier|title=Geometric methods in representation theory. II | pages=1–141| series=Sémin. Congr.|volume= 24-II|publisher= Soc. Math. France| location=Paris|year= 2012|arxiv=math/0611617|mr=3202707 |bibcode=2006math.....11617S}}