## Hall-Littlewood polynomials at roots of unity

Following [Mac, I (2.14)] and [Mac III (4.1)], for a partition $\lambda =\left({1}^{{m}_{1}}{2}^{{m}_{2}}\cdots \right)$ let

 ${z}_{\lambda }={1}^{{m}_{1}}{m}_{1}!{2}^{{m}_{2}}{m}_{2}!\cdots ,\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{z}_{\lambda }\left(t\right)={z}_{\lambda }\prod _{i\in {ℤ}_{>0}}\frac{1}{1-{t}^{{\lambda }_{i}}}$
As in [Mac III (2.11-2.13)] let
 $\phi \left(t\right)=\left(1-t\right)\left(1-{t}^{2}\right)\cdots \left(1-{t}^{\ell }\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{b}_{\lambda }\left(t\right)={\phi }_{{m}_{1}}\left(t\right){\phi }_{{m}_{2}}\left(t\right)\cdots$.
Then (see [Mac III (2.11-2.13)])
 ${Q}_{\lambda }\left(x;t\right)={b}_{\lambda }\left(t\right){P}_{\lambda }\left(x;t\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{q}_{\ell }\left(x;t\right)={Q}_{\left(\ell \right)}\left(x;t\right)$.
By [Mac III Ex. 7.1],
 ${Q}_{\left(\ell \right)}\left(x;t\right)=\sum _{\rho ⊢\ell }\frac{1}{{z}_{\rho }\left(t\right)}{p}_{\rho }\left(x\right)$,
and, by [Mac III Ex. 7.2],
 ${p}_{\ell }\left(x\right)=\sum _{\lambda ⊢\ell }{t}^{n\left(\lambda \right)}\left(\prod _{i=1}^{\ell \left(\lambda \right)-1}\left(1-{t}^{-i}\right)\right){P}_{\lambda }\left(x;t\right)$.

Let $r\in {ℤ}_{>0}$ and let $\zeta ={e}^{2\pi i/r}$. Let

 ${\Lambda }_{\left(r\right)}=ℤ\left[\zeta \right]\text{-span}\left\{{P}_{\lambda }\left(x;\zeta \right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{m}_{i}\left(\lambda \right).
By [Mac III Ex. 7.7], ${\Lambda }_{\left(r\right)}$ is a $ℤ\left[\zeta \right]$-algebra and
 ${\Lambda }_{\left(r\right)}{\otimes }_{ℤ\left[\zeta \right]}ℚ\left(\zeta \right)=ℚ\left(\zeta \right)\left[{p}_{i}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}i\ne 0\phantom{\rule{.5em}{0ex}}\mathrm{mod}\phantom{\rule{.5em}{0ex}}r\right]$.
By [To, Lemma 2.2] the map
 $\begin{array}{ccc}\frac{ℤ\left[{e}_{1},{e}_{2},\dots \right]}{⟨{e}_{i}\left({x}_{1}^{r},{x}_{2}^{r},\dots \right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}i\in {ℤ}_{>0}⟩}{\otimes }_{ℤ}ℤ\left[\zeta \right]& \stackrel{\sim }{⟶}& {\Lambda }_{\left(r\right)}\\ {e}_{i}& ⟼& {q}_{i}\end{array}$
is a $ℤ\left[\zeta \right]$-algebra isomorphism. By [To, Remark after Lemma 2.2],
 $\begin{array}{ccc}\frac{ℤ\left[{e}_{1},{e}_{2},\dots {e}_{k}\right]}{⟨{e}_{i}\left({x}_{1}^{r},{x}_{2}^{r},\dots \right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}i\in {ℤ}_{>0}⟩}{\otimes }_{ℤ}ℤ\left[\zeta \right]& \stackrel{\sim }{⟶}& \frac{{\Lambda }_{\left(r\right)}}{⟨{q}_{i}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}i>k⟩}\\ {e}_{i}& ⟼& {q}_{i}\end{array}$
is an isomorphism of $ℤ\left[\zeta \right]$-algebras and
 $\frac{{\Lambda }_{\left(r\right)}}{⟨{q}_{i}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}i>k⟩}$      has $ℤ\left[\zeta \right]$-bases
 $\left\{{Q}_{\lambda }\left(x;\zeta \right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{m}_{i}\left(\lambda \right)

In [FJ+, §3.1 eqn (6)] they define

 ${F}^{\left(r,2\right)}=\left\{f\in {ℂ\left[{x}_{1},\dots ,{x}_{k}\right]}^{{S}_{k}}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}f\left({x}_{1},\zeta {x}_{1},{\zeta }^{2}{x}_{1},\dots ,{\zeta }^{r-1}{x}_{1},{x}_{r+1},\dots ,{x}_{k}\right)=0\right\}.$
By [FJ+, Prop. 3.5],
 ${F}^{\left(r,2\right)}\simeq \frac{{\Lambda }_{\left(r\right)}}{⟨{q}_{i}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}i>k⟩}$
(is this exactly right??). The [FJ+] proof has two steps:
1. If ${m}_{i}\left(\lambda \right) for all $i$ then ${Q}_{\lambda }\left(x;\zeta \right)\in {F}^{\left(r,2\right)}$,
2. [FJ+, Lemma 3.2] which almost coincides with [To, Lemma 2.2] to say
 ${F}^{\left(r,2\right)}\simeq \frac{ℤ\left[{e}_{1},{e}_{2},\dots {e}_{k}\right]}{⟨{e}_{i}\left({x}_{1}^{r},{x}_{2}^{r},\dots \right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}i\in {ℤ}_{>0}⟩}{\otimes }_{ℤ}ℤ\left[\zeta \right]$
The fundamental formula is that
 $\text{if}\phantom{\rule{2em}{0ex}}Q\left(u\right)=\sum _{i\in {ℤ}_{>0}}{q}_{i}{u}^{i}\phantom{\rule{2em}{0ex}}\text{then}\phantom{\rule{2em}{0ex}}\prod _{j=0}^{r-1}Q\left({\zeta }^{j}u\right)=1,$
and if
 $E\left(u\right)=\sum _{i\in {ℤ}_{>0}}{e}_{i}{u}^{i}=\prod _{i\in {ℤ}_{>0}}\left(1+{x}_{i}u\right)$
then
 $\prod _{j=0}^{r-1}E\left({\zeta }^{j}u\right)=\sum _{\ell \in {ℤ}_{\ge 0}}{\left(-1\right)}^{\ell \left(r-1\right)}{e}_{\ell }\left({x}_{1}^{r},{x}_{1}^{r},\dots \right){u}^{\ell r}$.

## 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.

## References

[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?????.