Schur functions and multiplicities

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

Last update: 8 December 2012

Schur functions

Use notations for the Weyl group $W$ and the lattice $P$ as in Section 2. The group algebra of $P$ is the ring $$\mathbb{Z}\left[P\right]\text{with basis}\left\{X^{\lambda }\right|\lambda \in P\}\text{and product}{X}^{\lambda }{X}^{\mu }=X^{\lambda +\mu },$$ for $\lambda ,\mu \in P$. The group $W$ acts on $\mathbb{Z}\left[P\right]$ by $$w{X}^{\lambda }=X^{w\lambda },\text{for}w\in W,\lambda \in P.$$

The ring of symmetric functions and Fock space are

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

(5.1)

respectively. For $\lambda \in P$ define

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

(5.2)

The straightening laws for these elements are

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

(5.3)

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

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

(5.4)

where $P^{+}$ and $\rho$ are as in (2.14) and (2.16), respectively.

The Weyl characters or Schur functions are defined by

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

(5.5)

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

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

(5.6)

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 $$\sum _{\gamma \in P}{f}_{\gamma }{x}^{\gamma }=f=-s_{\alpha }f=\sum _{\gamma \in P}-f_{\gamma }{x}^{s_{\alpha }\gamma },\text{and so}f=\sum _{\genfrac{}{}{0pt}{}{\gamma \in P}{⟨\gamma ,{\alpha }^{\vee }⟩\ge 0}}{f}_{\gamma }(x^{\gamma }-x^{s_{\alpha }\gamma }),$$ 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 $$\text{each}f\in {\mathbb{Z}\left[P\right]}^{\det }\text{is divisible by}{x}^{\rho }\prod _{\alpha \in R^{+}}(1-x^{-\alpha })$$ (5.7) 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 $W$on $P$ is given by

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

(5.8)

The straightening law

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

(5.9)

for the Schur functions follows from the straightening law for the $a_{\mu }$ in (5.3).

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 $a_{\lambda +\rho }$, $$\begin{array}{rl}{\eta }_{\lambda }& =\left(\text{coefficient of}{s}_{\lambda }\text{in}f\right)=\left(\text{coefficient of}{a}_{\lambda +\rho }\text{in}f{a}_{\rho }\right)\\ & =\left(\text{coefficient of}{e}^{\lambda +\rho }\text{in}\sum _{\mu \in P}\sum _{w\in W}\det \right(w\left)f_{\mu }{e}^{\mu +w\rho }\right).\square \end{array}$$

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 $$s_{\lambda }\left(q^{{\rho }^{\vee }}\right)=\prod _{\alpha \in R^{+}}\frac{[⟨\lambda +\rho ,{\alpha }^{\vee }⟩]}{[⟨\rho ,{\alpha }^{\vee }⟩]}\text{and}{s}_{\lambda }\left(1\right)=\prod _{\alpha \in R^{+}}\frac{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{⟨\rho ,{\alpha }^{\vee }⟩}$$ where $\left[k\right]=(q^k-1)/(q-1)$ for an integer $k\ne 0$.

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

Multiplicities

The weight multiplicities are the integers $K_{\lambda \gamma }$, $\lambda \in P^{+}$, $\gamma \in P$, defined by the equations

$$s_{\lambda }=\sum _{\gamma \in P}{K}_{\lambda \gamma }{x}^{\gamma }=\sum _{\mu \in P^{+}}{K}_{\lambda \mu }{m}_{\mu }.$$

(5.10)

The tensor product multiplicities are the integers $c_{\mu \nu }^{\lambda }$, $\mu ,\nu ,\lambda \in P^{+}$, defined by the equations

$$s_{\mu }{s}_{\nu }=\sum _{\lambda \in P^{+}}{c}_{\mu \nu }^{\lambda }{s}_{\lambda }.$$

(5.11)

The partition function is the function $p:P\to {\mathbb{Z}}_{\ge 0}$ defined by the equation

$$\prod _{\alpha \in R^{+}}\frac{1}{1-x^{-\alpha }}=\sum _{\gamma \in P}p\left(\gamma \right)x^{-\gamma }.$$

(5.12)

Let $\lambda ,\mu ,\nu \in P^{+}$.

1. (a) $K_{\lambda \lambda }=1$,   $K_{\lambda ,w\mu }=K_{\lambda \mu }$, for $w\in W$,   and   $K_{\lambda \mu }=0$ unless $\mu \le \lambda$.
2. (b) $K_{\lambda \mu }=\sum _{w\in W}\det \left(w\right)p\left(w\right(\lambda +\rho )-(\mu +\rho \left)\right)$.
3. (c) $c_{\mu \nu }^{\lambda }=\sum _{v,w\in W}\det \left(vw\right)p\left(v\right(\mu +\rho )+w(\nu +\rho )-(\lambda +\rho )-\rho )$.

		Proof (a).
	The equality $K_{\lambda ,w\mu }=K_{\lambda \mu }$ follows from the definition and the fact that $s_{\lambda }\in {\mathbb{Z}\left[P\right]}^W$. If $w\ne 1$ then $w(\lambda +\rho )<\lambda +\rho$ so that $w(\lambda +\rho )-\rho <\lambda$ and $$s_{\lambda }=\left(\sum _{w\in W}\det \left(w\right)x^{w(\lambda +\rho )-\rho }\right)\cdot \prod _{\alpha \in R^{+}}\frac{1}{1-x^{-\alpha }}=x^{\lambda }+\left(\text{lower terms in dominance order}\right).$$ Thus $K_{\lambda \lambda =1}$and $K_{\lambda \mu }=0$ unless $\mu \le \lambda$. $\square$

		Proof (b).
	The coefficient of $x^{\mu }$ in $$s_{\lambda }=\left(\sum _{w\in W}\det \left(w\right)x^{w(\lambda +\rho )-\rho }\right)\prod _{\alpha \in R^{+}}\frac{1}{1-x^{-\alpha }}=\sum _{\genfrac{}{}{0pt}{}{w\in W}{\gamma \in Q^{+}}}\det \left(w\right)p\left(\gamma \right)x^{w(\lambda +\rho )-\rho -\gamma },$$ has a contribution $\det \left(w\right)p\left(\gamma \right)$ when $w(\lambda +\rho )-\rho -\gamma =\mu$ so that $\gamma =w(\lambda +\rho )-(\mu +\rho )$. $\square$

		Proof (c).
	Let $\epsilon =\sum _{w\in W}\det \left(w\right)w$. Since $c_{\lambda }^{\mu \nu }$ is the coefficient of $x^{\nu +\rho }$ in $$\begin{array}{rcl}{s}_{\mu }{s}_{\nu }{a}_{\rho }& =& \frac{\epsilon \left(x^{\mu +\rho }\right)\epsilon \left(x^{\nu +\rho }\right)}{a_{\rho }}=\left(\sum _{v,w\in W}\det \left(vw\right)x^{v(\mu +\rho )+w(\nu +\rho )-\rho }\right)\left(\prod _{\alpha \in R^{+}}\frac{1}{1-x^{-\alpha }}\right)\\ & =& \sum _{\genfrac{}{}{0pt}{}{v,w\in W}{\gamma \in Q^{+}}}\det \left(vw\right)p\left(\gamma \right)x^{v(\mu +\rho )+w(\nu +\rho )-\gamma -\rho },\end{array}$$ there is a contribution $\det \left(vw\right)p\left(\gamma \right)$ to the coefficient $c_{\mu \nu }^{\lambda }$ when $\lambda +\rho =v(\mu +\rho )+w(\nu +\rho )-\gamma -\rho$ so that $\gamma =v(\mu +\rho )+w(\mu +\rho )-(\lambda +\rho )-\rho$. $\square$

Fix $J\subseteq \{1,2,\dots ,n\}$. The subgroup of $W$ generated by the reflections in the hyperplanes $H_{{\alpha }_j}$, $j\in J$,

$$W_J=⟨s_j|j\in J⟩\text{acts on}{𝔥}_{ℝ}^{*}\text{ with }{C}_J=\left\{\mu \in {𝔥}_{ℝ}^{*}\right|⟨\mu ,{\alpha }_j^{\vee }⟩>0\text{for}j\in J\}$$

(5.13)

as a fundamental chamber. The group $W_J$ acts on $P$ and

$${\mathbb{Z}\left[P\right]}^{W_J}=\left\{f\in \mathbb{Z}\right[P\left]\right|wf=f\text{for}w\in W_J\}$$

(5.14)

is a subalgebra of $\mathbb{Z}\left[P\right]$ which contains ${\mathbb{Z}\left[P\right]}^W$. If

$$\overline{C_J}=\left\{\mu \in {𝔥}_{ℝ}^{*}\right|⟨\mu ,{\alpha }_j^{\vee }⟩\ge 0\text{for}j\in J\},P_{+}^J=P\cap \overline{C_J},{\rho }_J=\sum _{j\in J}{\omega }_j,$$

(5.15)

$$a_J^{\mu }=\sum _{w\in W_J}\det \left(w\right)w{X}^{\mu },\text{for}\mu \in P,\text{and}{s}_{\lambda }^J=\frac{a_{\lambda +{\rho }_J}^J}{a_{{\rho }_J}^J},\text{for}\lambda \in P,$$

(5.16)

then $$\left\{s_{\lambda }^J\right|\lambda \in P_J^{+}\}\text{is a basis of}{\mathbb{Z}\left[P\right]}^{W_J}.$$ The restriction multiplicities are the integers $c_{J,\nu }^{\lambda }$ given by

$$s_{\lambda }=\sum _{\nu \in P_J^{+}}{c}_{J,\nu }^{\lambda }{s}_{\nu }^J.$$

(5.17)

Notes and References

These notes are intended to supplement various lecture series given by Arun Ram. This is a typed version of section 5.1 of the paper [Ram2006].

References

[Ram2006] Alcove walks, Hecke algebras, Spherical functions, crystals and column strict tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (Special Issue: In honor of Robert MacPherson, Part 2 of 3) (2006) 963-1013. MR2282411

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