Multiplicities

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: 22 January 2009

Multiplicities

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

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

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

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

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

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

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

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

		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 $$\begin{array}{rcl}{s}_{\lambda }& =& \left(\sum _{w\in W}\det \left(w\right)x^{w(\lambda +\rho )-\rho }\right).\prod \alpha \in R^{+}\frac{1}{1-x^{-\alpha }}\\ & =& x^{\lambda }+\left(\text{lower terms in dominance order}\right).\end{array}$$ Thus $K_{\lambda \lambda =1}$ and $K_{\lambda \mu }=0$ unless $\mu \le \lambda \text{.}\square$

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

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

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

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

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

is a subalgebra of $\mathbb{Z}\left[P\right]$ which contains ${\mathbb{Z}\left[P\right]}^W.$ If $$\overline{C_J}=\left\{\mu \in {𝔥}_{ℝ}^{*}|⟨\mu ,{\alpha }_j^{\vee }⟩\ge 0\text{ for }j\in J\right\},$$

$$P_{+}^J=P\cap \overline{C_J},{\rho }_J=\sum _{j\in J}{\omega }_j,$$

$$a_J^{\mu }=\sum _{w\in W_J}\det \left(w\right)w{X}^{\mu },\text{ for }\mu \in P,\text{ and}$$

$$s_{\lambda }^J=\frac{a_{\lambda +{\rho }_j}^J}{a_{{\rho }_J}^J},\text{ for}\lambda \in P,$$

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

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

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)