Schur functions

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 18 January 2009

Schur functions

We use the denotations W for the Weyl group and P for the lattice as in Section 2 [ REFERENCE TO OTHER DOCUMENT ] . The group algebra of P is the ring

for $\lambda \text{,}\mu \in P.$ The group $W$ acts on $\mathbb{Z}\left[P\right]$ by $$w{X}^{\lambda }=X^{w\lambda }\text{,}\text{for}w\in W\text{, }\lambda \in P\text{.}$$

The ring of symmetric functions and Fock space are

$$\begin{array}{c}{\mathbb{Z}\left[P\right]}^W=\left\{f\in \mathbb{Z}\left[P\right]|wf=f\text{ for all }w\in W\right\}\text{and}\\ {\mathbb{Z}\left[P\right]}^{\det }=\left\{f\in \mathbb{Z}\left[P\right]|wf=\det (w)f\text{ for all }w\in W\right\},\end{array}$$

respectively. For $\lambda \in P$ define

$$\begin{array}{c}{m}_{\lambda }=\sum _{\gamma \in W_{\lambda }}{X}^{\gamma }\text{and}{a}_{\lambda }=\sum _{w\in W}\det \left(w\right)X^{w\lambda }.\end{array}$$

The straightening laws for these elements are

$$m_{w\lambda }=m_{\lambda }\text{and}{a}_{w\lambda }=\det \left(w\right)a_{\lambda }\text{, for}w\in W\text{ and }\lambda \in P.$$

The second relation implies that $a_{\lambda }=0$ if there exists $w\in W_{\lambda }$ with $\det \left(w\right)\ne 1$, and it follows from the straightening laws that

$${\mathbb{Z}\left[P\right]}^W\text{has basis }\left\{m_{\lambda }|\lambda \in P^{+}\right\}\text{and}{\mathbb{Z}\left[P\right]}^{\det }\text{ has basis }\left\{a_{\lambda +\rho }|\lambda \in P^{+}\right\}$$

where $M^{+}\text{and}\rho$ are as in (2.14) and (2.16) [ REFERENCE TO OTHER MATERIAL ] respectively.

The Weyl characters or Schur functions are defined by

$$s_{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }},\text{ for }\lambda \in P.$$

The following theorem shows that the $s_{\lambda }$ are the elements of $\mathbb{Z}\left[P\right]$ and that

$\begin{array}{rcl}{\mathbb{Z}\left[P\right]}^W& \text{has basis}& \left\{s_{\lambda }|\lambda \in P^{+}\right\}.\end{array}$

Fock space ${\mathbb{Z}\left[P\right]}^{\det }$ is a free ${\mathbb{Z}\left[P\right]}^W$ module with generator

$$a_{\rho }=x^{\rho }\prod _{\alpha \in R^{+}}(1-x^{-\alpha })\text{ and the map }\begin{array}{rcl}{\mathbb{Z}\left[P\right]}^W& \to & {\mathbb{Z}\left[P\right]}^{\det }\\ f& \mapsto & a_{\rho }f\\ s_{\lambda }& \mapsto & a_{\lambda +\rho }\end{array}$$

is a ${\mathbb{Z}\left[P\right]}^W$ module isomorphism.

		Proof.
	Let $f\in {\mathbb{Z}\left[P\right]}^{\det }$ and let $\alpha \in R^{+}.$ If $f_{\gamma }$ is the coefficient of $x^{\gamma }$ in $f$ then $$\sum _{\gamma \in P}{f}_{\gamma }{x}^{\gamma }=f=-s_{\alpha }f=\sum _{\gamma \in P}-f_{\gamma }{x}^{s_{\alpha }\gamma },$$    and so    $$f=\sum _{\gamma \in P,⟨\gamma ,{\alpha }^{\vee }⟩\ge 0}{f}_{\gamma }(x^{\gamma }-x^{s_{\alpha }\gamma }),$$ since $f_{s_{\alpha }\gamma }=-f_{\gamma }.$ Since each term $x^{\gamma }-x^{s_{\alpha }\gamma }$ is divisible by $1-x^{-\alpha }$, $f$ is divisible by $1-x^{-\alpha }$, and thus $$\text{each }f\in {\mathbb{Z}\left[P\right]}^{\det }\text{ is divisible by }{x}^{\rho }\prod _{\alpha \in R^{+}}(1-x^{-\alpha })$$ since the polynomials $1-x^{-\alpha }, \alpha \in R^{+}$ are coprime in $\mathbb{Z}\left[P\right]$ and $x^{\rho }$ is a unit in $\mathbb{Z}\left[P\right]$. Comparing coefficients of the maximal terms in $a_{\rho }$ and $x^{\rho }\prod _{\alpha \in R^{+}}(1-x^{-\alpha })$ shows that $$a_{\rho }=x^{\rho }\prod _{\alpha \in R^{+}}(1-x^{-\alpha }).$$ Thus each $f\in {\mathbb{Z}\left[P\right]}^{\det }$ is divisible by $a_{\rho }$ and so the inverse of multiplication by $a_{\rho }$ is well defined. $\square$

The dot action of $S_n$ on $P$ is given by

$$w\circ \mu =w(\mu +\rho )-\rho ,\text{ for }w\in S_n,\mu \in P.$$

The straightening law

$$s_{w\circ \mu }=\det \left(w\right)s_{\mu }\text{, for }\mu \in P,w\in W$$

for the Schur functions follows from the straightening law for the $a_{\mu }$ in (1.3).

Let $f\in {\mathbb{Z}\left[P\right]}^W$ and write $f=\sum _{\gamma }{f}_{\gamma }{x}^{\gamma }$ so that $f_{\gamma }$ is the coefficient of $x^{\gamma }$ in $f$. Then

$$f=\sum _{\mu \in P^{+}}{f}_{\mu }{m}_{\mu }=\sum _{\lambda \in P^{+}}{\eta }^{\lambda }{s}_{\lambda },\text{ where }{\eta }^{\lambda }=\sum _{w\in W}\det \left(w\right)f_{\lambda +\rho -w\rho }.$$

		Proof.
	The first equality is immediate from the definition of $m_{\mu }.$ Since $f\in {\mathbb{Z}\left[P\right]}^W$ and the $s_{\lambda },\lambda \in P^{+},$ are a basis of ${\mathbb{Z}\left[P\right]}^W$, the element $f$ can be written as a linear combination of $s_{\lambda }$. Then, since $e^{\lambda +\rho }$ is the unique dominant term in $a_{\lambda +\rho },$ ${\eta }_{\lambda }=\left(\text{coefficient of}{s}_{\lambda }\text{ in }f\right)=\left(\text{coefficient of }{a}_{\lambda +\rho }\text{ in }f{a}_{\rho }\right)=\left(\text{coefficient of }{e}^{\lambda +\rho }\text{ in }\sum _{\mu \in P}\sum _{w\in W}\det \right(w\left)f_{\mu }{e}^{\mu +w\rho }\right)\text{.}\square$

If $\nu \in {𝔥}_{ℝ}^{*}$ and $f=\sum _{\mu \in P}{f}_{\mu }{e}^{\mu }\in \mathbb{Z}\left[P\right]$ define $f\left(e^{\nu }\right)=\sum _{\mu \in P}{f}_{\mu }{e}^{⟨\mu ,\nu ⟩}.$ Let $\lambda \in P^{+},t\in {ℝ}_{>0},q=e^t$ and ${\rho }^{\vee }=\frac{1}{2}\sum _{\alpha \in R^{+}}{\alpha }^{\vee }.$ Then $$s_{\lambda }\left(q^{{\rho }^{\vee }}\right)=\prod _{\alpha \in R^{+}}\frac{\left[⟨\lambda +\rho ,{\alpha }^{\vee }⟩\right]}{\left[⟨\rho ,{\alpha }^{\vee }⟩\right]}\text{ and }{s}_{\lambda }\left(1\right)=\prod _{\alpha \in R^{+}}\frac{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{⟨\rho ,{\alpha }^{\vee }⟩}$$   where   $\left[k\right]=\left(q^k-1\right)/\left(q-1\right)\text{ for an integer }k\ne 0.$

		Proof.
	Thus $s_{\lambda }\left(e^{t{\rho }^{\vee }}\right)=\frac{a_{\lambda +\rho }\left(e^{t{\rho }^{\vee }}\right)}{a_{\rho }\left(e^{t{\rho }^{\vee }}\right)}=\frac{e^{⟨{\rho }^{\vee },t\left(\lambda +\rho \right)⟩}}{e^{⟨{\rho }^{\vee },t\rho ⟩}}\prod _{\alpha \in R^{+}}\frac{1-e^{⟨-{\alpha }^{\vee },t,\left(\lambda +\rho \right)⟩}}{1-e^{⟨-{\alpha }^{\vee },t,\rho ⟩}}=q^{-⟨\lambda ,{\rho }^{\vee }⟩}\prod _{\alpha \in R^{+}}\frac{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{⟨\rho ,{\alpha }^{\vee }⟩}.$ and $$s_{\lambda }\left(1\right)=\underset{q\to 1}{\lim }{s}_{\lambda }\left(q^{{\rho }^{\vee }}\right)=\prod _{\alpha \in R^{+}}\frac{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{⟨\rho ,{\alpha }^{\vee }⟩}.\square$$$$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)