Hall-Littlewood polynomials at roots of unity

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $\lambda =(1^{m_1}{2}^{m_2}\cdots )$ let

$z_{\lambda }=1^{m_1}{m}_1!2^{m_2}{m}_2!\cdots ,\text{and}{z}_{\lambda }\left(t\right)=z_{\lambda }{\displaystyle \prod _{i\in {\mathbb{Z}}_{>0}}\frac{1}{1-t^{{\lambda }_i}}}$

As in [Mac III (2.11-2.13)] let

$\phi \left(t\right)=(1-t)(1-t^2)\cdots (1-t^{\ell })\text{and}{b}_{\lambda }\left(t\right)={\phi }_{m_1}\left(t\right){\phi }_{m_2}\left(t\right)\cdots$.

Then (see [Mac III (2.11-2.13)])

$Q_{\lambda }(x;t)=b_{\lambda }\left(t\right)P_{\lambda }(x;t)\text{and}{q}_{\ell }(x;t)=Q_{\left(\ell \right)}(x;t)$.

By [Mac III Ex. 7.1],

$Q_{\left(\ell \right)}(x;t)=\sum _{\rho ⊢\ell }\frac{1}{z_{\rho }\left(t\right)}{p}_{\rho }\left(x\right)$,

and, by [Mac III Ex. 7.2],

$p_{\ell }\left(x\right)=\sum _{\lambda ⊢\ell }{t}^{n\left(\lambda \right)}\left(\prod _{i=1}^{\ell \left(\lambda \right)-1}(1-t^{-i})\right)P_{\lambda }(x;t)$.

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

${\Lambda }_{\left(r\right)}=\mathbb{Z}\left[\zeta \right]\text{-span}\left\{P_{\lambda }\right(x;\zeta \left)\right|m_i\left(\lambda \right)<r\}$.

By [Mac III Ex. 7.7], ${\Lambda }_{\left(r\right)}$ is a $\mathbb{Z}\left[\zeta \right]$-algebra and

${\Lambda }_{\left(r\right)}{\otimes }_{\mathbb{Z}\left[\zeta \right]}\mathbb{Q}\left(\zeta \right)=\mathbb{Q}\left(\zeta \right)\left[p_i\right|i\ne 0\mathrm{mod}r]$.

By [To, Lemma 2.2] the map

$\begin{array}{ccc}{\displaystyle \frac{\mathbb{Z}[e_1,e_2,\dots ]}{⟨e_i(x_1^r,x_2^r,\dots )|i\in {\mathbb{Z}}_{>0}⟩}{\otimes }_{\mathbb{Z}}\mathbb{Z}\left[\zeta \right]}& \stackrel{\sim }{⟶}& {\Lambda }_{\left(r\right)}\\ e_i& ⟼& q_i\end{array}$

is a $\mathbb{Z}\left[\zeta \right]$-algebra isomorphism. By [To, Remark after Lemma 2.2],

$\begin{array}{ccc}{\displaystyle \frac{\mathbb{Z}[e_1,e_2,\dots e_k]}{⟨e_i(x_1^r,x_2^r,\dots )|i\in {\mathbb{Z}}_{>0}⟩}{\otimes }_{\mathbb{Z}}\mathbb{Z}\left[\zeta \right]}& \stackrel{\sim }{⟶}& {\displaystyle \frac{{\Lambda }_{\left(r\right)}}{⟨q_i|i>k⟩}}\\ e_i& ⟼& q_i\end{array}$

is an isomorphism of $\mathbb{Z}\left[\zeta \right]$-algebras and

${\displaystyle \frac{{\Lambda }_{\left(r\right)}}{⟨q_i|i>k⟩}}$      has $\mathbb{Z}\left[\zeta \right]$-bases

$\left\{Q_{\lambda }\right(x;\zeta \left)\right|m_i\left(\lambda \right)<r\text{and}{\lambda }_1\le k\left\}\text{and}\right\{q_{\lambda }(x;\zeta )\left|m_i\right(\lambda )<r\text{and}{\lambda }_1\le k\}.$

In [FJ+, §3.1 eqn (6)] they define

$F^{(r,2)}=\{f\in {ℂ[x_1,\dots ,x_k]}^{S_k}|f(x_1,\zeta x_1,{\zeta }^2{x}_1,\dots ,{\zeta }^{r-1}{x}_1,x_{r+1},\dots ,x_k)=0\}.$

By [FJ+, Prop. 3.5],

$F^{(r,2)}\simeq {\displaystyle \frac{{\Lambda }_{\left(r\right)}}{⟨q_i|i>k⟩}}$

(is this exactly right??). The [FJ+] proof has two steps:

1. If $m_i\left(\lambda \right)<r$ for all $i$ then $Q_{\lambda }(x;\zeta )\in F^{(r,2)}$,
2. [FJ+, Lemma 3.2] which almost coincides with [To, Lemma 2.2] to say

   $F^{(r,2)}\simeq {\displaystyle \frac{\mathbb{Z}[e_1,e_2,\dots e_k]}{⟨e_i(x_1^r,x_2^r,\dots )|i\in {\mathbb{Z}}_{>0}⟩}{\otimes }_{\mathbb{Z}}\mathbb{Z}\left[\zeta \right]}$

The fundamental formula is that

$\text{if}Q\left(u\right)=\sum _{i\in {\mathbb{Z}}_{>0}}{q}_i{u}^i\text{then}\prod _{j=0}^{r-1}Q\left({\zeta }^{j}u\right)=1,$

and if

$E\left(u\right)=\sum _{i\in {\mathbb{Z}}_{>0}}{e}_i{u}^i=\prod _{i\in {\mathbb{Z}}_{>0}}(1+x_{i}u)$

then

$\prod _{j=0}^{r-1}E\left({\zeta }^{j}u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{\left(-1\right)}^{\ell (r-1)}{e}_{\ell }(x_1^r,x_1^r,\dots )u^{\ell r}$.

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?????.

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