Hall-Littlewood polynomials at roots of unity

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 1 June 2011

Hall-Littlewood polynomials at roots of unity

Following [Mac, I (2.14)] and [Mac III (4.1)], for a partition λ=( 1m1 2m2 ) let

zλ = 1m1 m1! 2m2 m2! , and zλ(t) =zλ i>0 11-tλi
As in [Mac III (2.11-2.13)] let
φ(t) =(1-t) (1-t2) (1-t) and bλ(t) = φm1(t) φm2(t) .
Then (see [Mac III (2.11-2.13)])
Qλ(x;t) = bλ(t) Pλ(x;t) and q(x;t) = Q() (x;t) .
By [Mac III Ex. 7.1],
Q() (x;t) = ρ 1zρ(t) pρ(x) ,
and, by [Mac III Ex. 7.2],
p(x) = λ tn(λ) ( i=1 (λ)-1 (1-t-i) ) Pλ(x;t) .

Let r>0 and let ζ= e2πi/r. Let

Λ(r) = [ζ]-span {Pλ(x;ζ) | mi(λ) <r} .
By [Mac III Ex. 7.7], Λ(r) is a [ζ] -algebra and
Λ(r) [ζ] (ζ) = (ζ) [pi | i0mod r] .
By [To, Lemma 2.2] the map
[e1, e2,] ei( x1r, x2r, ) | i>0 [ζ] Λ(r) ei qi
is a [ζ]-algebra isomorphism. By [To, Remark after Lemma 2.2],
[e1, e2,ek] ei( x1r, x2r, ) | i>0 [ζ] Λ(r) qi | i>k ei qi
is an isomorphism of [ζ]-algebras and
Λ(r) qi | i>k      has [ζ]-bases
{Qλ(x;ζ) | mi(λ)<r and λ1k} and {qλ(x;ζ) | mi(λ)<r and λ1k}.

In [FJ+, §3.1 eqn (6)] they define

F(r,2) = { f [x1, ,xk] Sk | f(x1, ζx1, ζ2x1, , ζr-1 x1, xr+1, ,xk )=0}.
By [FJ+, Prop. 3.5],
F(r,2) Λ(r) qi | i>k
(is this exactly right??). The [FJ+] proof has two steps:
  1. If mi(λ) <r for all i then Qλ(x;ζ) F(r,2) ,
  2. [FJ+, Lemma 3.2] which almost coincides with [To, Lemma 2.2] to say
    F(r,2) [e1, e2,ek] ei( x1r, x2r, ) | i>0 [ζ]
The fundamental formula is that
if Q(u) = i>0 qiui then j=0 r-1 Q(ζju) =1,
and if
E(u) = i>0 eiui = i>0 (1+xiu)
j=0 r-1 E(ζju) = 0 (-1) (r-1) e( x1r, x1r, ) ur .

Notes and References

  1. The bulk of the paper of Morris is contained in [Mac III Ex. 7.7]. In the last section Morris makes a very interesting comparison to the modular representation theory of the symmetric group.
  2. [Mac II Ex. 5.7] gives a "factorisation formula" due to [LLT].
  3. These notes are a typed version of a scan of handwritten notes sent to Z. Daugherty on 16 March 2011.


[FJ+] B. Feigin, M. Jimbo, T. Miwa, E. Mukhin and Y. Takeyama, Symmetric polynomials vanishing on the diagonals shifted by roots of unity, arXiv:math/0209126 [math.QA], Int Math. res. Notices 2003 no. 18, 1015-1034.

[Mac] I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford Univ. Press 1995. MR?????.

[Mo] A.O. Morris, On an algebra of symmetric functions, Quart. J. Math. 16 (1965), 53-64. MR?????.

[To] B. Totaro, Towards a Schubert calculus for complex reflection groups, Math. Proc. Camb. Phil. Soc. 134 (2003) 83-93, http://www.dpmms.cam.ac.uk/~bt219/papers.html MR?????.

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