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.

Contents

Construction

References

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.