Symmetric and alternating functions and their $q$-analogues

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

Last update: 26 September 2012

Symmetric and alternating functions and their $q$-analogues

Let $1_0$ and ${\epsilon }_0$ be the elements of the finite Hecke algebra $H$ which are determined by

$$\begin{array}{cccc}{1}_0^2=1_0& \text{and}& T_i{1}_0=q{1}_0,& \text{for all}1\le i\le n,\\ {\epsilon }_0^2={\epsilon }_0& \text{and}& T_i{\epsilon }_0=(-q^{-1}){\epsilon }_0,& \text{for all}1\le i\le n\text{.}\end{array}$$

In terms of the basis $\{T_w\mid w\in W\}$ of $H$ these elements have the explicit formulae

$$\begin{array}{cc}{1}_0=\frac{1}{W_0\left(q^2\right)}\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w,\text{and}{\epsilon }_0=\frac{1}{W_0\left(q^{-2}\right)}\sum _{w\in W}{(-q)}^{-\ell \left(w\right)}{T}_w,& \text{(2.1)}\end{array}$$

where $W_0\left(t\right)=\sum _{w\in W}{t}^{\ell \left(w\right)}\text{.}$ (To define these elements one should adjoin the element $W_0{\left(q^2\right)}^{-1}$ to $𝕂$ or to $H\text{.)}$ The elements $1_0$ and ${\epsilon }_0$ are $q$-analogues of the elements in the group algebra of $W$ given by

$$\begin{array}{cc}1=\frac{1}{\mid W\mid }\sum _{w\in W}w\text{and}\epsilon =\frac{1}{\mid W\mid }\sum _{w\in W}{(-1)}^{\ell \left(w\right)}w,& \text{(2.2)}\end{array}$$

and the vector spaces $1_0\stackrel{\sim }{H}{1}_0$ and ${\epsilon }_0\stackrel{\sim }{H}{1}_0$ are $q$-analogues of the vector spaces (more precisely, free $𝕂=\mathbb{Z}[q,q^{-1}]$-modules) of symmetric functions and alternating functions,

$$\begin{array}{cc}\begin{array}{ccc}𝕂{\left[P\right]}^W& =& \{f\in 𝕂\left[P\right]\mid wf=f\text{for all}w\in W\}=1𝕂\left[P\right],\\ 𝒜& =& \{f\in 𝕂\left[P\right]\mid wf={(-1)}^{\ell \left(w\right)}f\text{for all}w\in W\}=\epsilon 𝕂\left[P\right],\end{array}& \text{(2.3)}\end{array}$$

respectively, where the action of $W$ on $𝕂\left[P\right]$ is as defined in 1.29.

For $\mu \in P$ let the orbit $W_{\mu }$ and the stabilizer $W_{\mu }$ of $\mu$ be defined by

$$W_{\mu }=\{w\mu \mid w\in W\}\text{and}{W}_{\mu }=\{w\in W\mid w\mu =\mu \}\text{.}$$

Then define

$$\begin{array}{cc}\begin{array}{cc}{m}_{\mu }=\sum _{\gamma \in W_{\mu }}{x}^{\gamma }=\frac{\mid W\mid }{\mid W_{\mu }\mid }1x^{\mu },& a_{\mu }=\sum _{w\in W}{\left(-1\right)}^{\ell \left(w\right)}w{x}^{\mu }=\mid W\mid \epsilon x^{\mu },\\ M_{\mu }=1_0{x}^{\mu }{1}_0,& A_{\mu }={\epsilon }_0{x}^{\mu }{1}_0\text{.}\end{array}& \text{(2.4)}\end{array}$$

Theorem 2.2 below shows that the elements in (2.4) which are indexed by elements of $P^{+}$ and $P^{++}$ form bases (over $𝕂\text{)}$ of $𝕂{\left[P\right]}^W,$ $𝒜,$ $1_0\stackrel{\sim }{H}{1}_0,$ and ${\epsilon }_0\stackrel{\sim }{H}{1}_0\text{.}$ This will be a consequence of the following straightening rules. The straightening law for the $M_{\mu }$ given in the following Proposition is a generalization of [Mac1995, III §2 Ex. 2].

For $\gamma \in P$ let $m_{\gamma },$ $a_{\gamma },$ $M_{\gamma },$ and $A_{\gamma }$ be as defined in (2.4). Let ${\alpha }_i$ be a simple root and let $\mu \in P$ be such that $d=⟨\mu ,{\alpha }_i^{\vee }⟩\ge 0\text{.}$ Then

$$m_{s_i\mu }=m_{\mu },a_{s_i\mu }=-a_{\mu },\text{and}{A}_{s_i\mu }=-A_{\mu }\text{.}$$

Letting $t=q^{-2},$ $M_{\mu }=M_{s_i\mu }$ if $d=0,$ and if $d>0$ then

$$\begin{array}{ccccc}{M}_{s_i\mu }& =& t{M}_{\mu }& +& \left(\sum _{j=1}^{\lfloor d/2-1\rfloor }(t^2-1)t^{j-1}{M}_{\mu -j{\alpha }_i}\right)\\ & & & +& \{\begin{array}{cc}(t-1)t^{d/2-1}{M}_{\mu -(d/2){\alpha }_i},& \text{if}d\text{is even,}\\ 0,& \text{if}d\text{is odd.}\end{array}\end{array}$$

		Proof.
	The first two equalities follow from the definitions of $m_{\lambda }$ and $a_{\mu }$ and the fact that $\ell \left(s_i\right)=1\text{.}$ Let $\mu \in P$ such that $d=⟨\mu ,{\alpha }_i^{\vee }⟩\ge 0\text{.}$ Since $x^{\mu }+x^{s_i\mu }$ is in the center of the tiny little affine Hecke algebra generataed by $T_i$ and the $x^{\gamma },$ $\gamma \in P,$ $$\begin{array}{ccc}{A}_{\mu }+A_{s_i\mu }& =& {\epsilon }_0(x^{\mu }+x^{s_i\mu })1_0=q^{-1}{\epsilon }_0(x^{\mu }+x^{s_i\mu })T_i{1}_0\\ & =& q^{-1}{\epsilon }_0{T}_i(x^{\mu }+x^{s_i\mu })1_0=-q^{-2}{\epsilon }_0(x^{\mu }+x^{s_i\mu })1_0\\ & =& -q^{-2}(A_{\mu }+A_{s_i\mu })\text{.}\end{array}$$ Thus $A_{\mu }+A_{s_i\mu }=0$ which establishes the third statement. If $d=0$ then, by definition, $M_{\mu }=M_{s_i\mu }\text{.}$ If $d>0$ then multiplying the fourth relation in (1.21) by $1_0$ on both the left and the right (and then multiplying by $q^{-1}\text{)}$ gives $$1_0(x^{s_i\mu }-x^{\mu })1_0=q^{-1}(q-q^{-1})1_0\left(\frac{x^{s_i\mu }-x^{\mu }}{1-x^{-{\alpha }_i}}\right)1_0\text{.}$$ Subtracting the same relation with $\mu$ replaced by $\mu -{\alpha }_i$ gives $$\begin{array}{ccc}{1}_0(x^{s_i\mu }-x^{\mu })1_0-1_0(x^{s_i\mu +{\alpha }_i}-x^{\mu -{\alpha }_i})1_0& =& (1-q^{-2})1_0\left(\frac{x^{s_i\mu }-x^{\mu }-x^{s_i\mu +{\alpha }_i}+x^{\mu -{\alpha }_i}}{1-x^{-{\alpha }_i}}\right)1_0\\ & =& (1-q^{-2})1_0(-x^{s_i\mu +{\alpha }_i}-x^{\mu })1_0\text{.}\end{array}$$ So $$1_0{x}^{s_i\mu }{1}_0=q^{-2}{1}_0{x}^{\mu }{1}_0-1_0{x}^{\mu -{\alpha }_i}{1}_0+q^{-2}{1}_0{x}^{s_i\mu +{\alpha }_i}{1}_0\text{.}$$ Inductively applying this relation yields the result. The first cases are $$M_{s_i\mu }=\{\begin{array}{cc}{M}_{\mu },& \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=0,\\ q^{-2}{M}_{\mu },& \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=1,\\ q^{-2}{M}_{\mu }+(q^{-2}-1)M_{\mu -{\alpha }_i},& \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=2,\\ q^{-2}{M}_{\mu }+(q^{-4}-1)M_{\mu -{\alpha }_i},& \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=3,\\ q^{-2}{M}_{\mu }+(q^{-4}-1)M_{\mu -{\alpha }_i}+q^{-2}(q^{-2}-1)M_{\mu -2{\alpha }_i},& \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=4\text{.}\end{array}$$ $\square$

Proposition 2.1 implies that, for all $\mu \in P$ and $w\in W,$

$$\begin{array}{cc}{m}_{w\mu }=m_{\mu },a_{w\mu }={(-1)}^{\ell \left(w\right)}{a}_{\mu },\text{and}{A}_{w\mu }{(-1)}^{\ell \left(w\right)}{A}_{\mu }\text{.}& \text{(2.5)}\end{array}$$

Let $𝕂=\mathbb{Z}[q,q^{-1}]\text{.}$ As free $𝕂$-modules

$$\begin{array}{cccc}𝕂{\left[P\right]}^W& \text{has basis}\{m_{\lambda }\mid \lambda \in P^{+}\},& 1_0\stackrel{\sim }{H}{1}_0& \text{has basis}\{M_{\lambda }\mid \lambda \in P^{+}\},\\ 𝒜& \text{has basis}\{a_{\mu }\mid \mu \in P^{++}\},& {\epsilon }_0\stackrel{\sim }{H}{1}_0& \text{has basis}\{A_{\mu }\mid \mu \in P^{++}\}\text{.}\end{array}$$

		Proof.
	Since $\{x^{\mu }{T}_w\mid \mu \in P,w\in W\}$ form a basis of $\stackrel{\sim }{H}$ the elements $M_{\mu }=1_0{x}^{\mu }{1}_0=q^{-\ell \left(w\right)}{1}_0{x}^{\mu }{T}_w{1}_0,$ $\mu \in P,$ span $1_0\stackrel{\sim }{H}{1}_0\text{.}$ By Proposition 2.1, if $\mu$ is on the negative side of a hyperplane $H_{{\alpha }_i},$ that is, if $⟨\mu ,{\alpha }_i^{\vee }⟩<0,$ then $M_{\mu }$ can be rewritten as a linear combination of $M_{\gamma }$ such that all terms have $\gamma$ on the nonnegative side of $H_{{\alpha }_i}\text{.}$ By repeatedly applying the relation in Proposition 2.1, $M_{\mu }$ can be rewritten as a linear combination of $M_{\gamma }$ such that all terms have $\gamma$ on the nonnegative side of $H_{{\alpha }_1},\dots ,H_{{\alpha }_n},$ that is, $\gamma \in P^{+}=P\cap \bar{C},$ where $\bar{C}=\{x\in {ℝ}^n\mid ⟨x,{\alpha }_i^{\vee }⟩\ge 0\text{for all}1\le i\le n\}\text{.}$ If $\lambda \in P^{},$ using the fourth relation in (1.21), $$\begin{array}{ccc}{M}_{\lambda }& =& 1_0{x}^{\lambda }{1}_0=\frac{1}{W_0\left(q^2\right)}\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_w{x}^{\lambda }{1}_0=\frac{1}{W_0\left(q^2\right)}\sum _{\gamma ,v,w}{q}^{\ell \left(w\right)}{d}_{v,\gamma }{x}^{\gamma }{T}_v{1}_0\\ & =& \frac{1}{W_0\left(q^2\right)}\sum _{\gamma ,v,w}{q}^{\ell \left(w\right)}{d}_{v,\gamma }{x}^{\gamma }{q}^{\ell \left(v\right)}{1}_0=\frac{1}{W_0\left(q^2\right)}\sum _{\gamma }{d}_{\gamma }{x}^{\gamma }{1}_0,\end{array}$$ where $d_{v,\gamma }$ and $d_{\gamma }$ are some polynomials in $\mathbb{Z}[q,(q-q^{-1})]$ such that $d_{v,v\lambda }=1$ so that $d_{w_0\lambda }=1\text{.}$ Furthermore $d_{\gamma }=0$ unless $\gamma$ is in the convex hull of the points in the orbit $W\lambda \text{.}$ Thus the coefficient of $x^{w_0\lambda }$ in $M_{\lambda }$ is $W_0{\left(q^2\right)}^{-1}{q}^{2\ell \left(w0\right)}$ and the coefficient of $x^{\gamma }{T}_v$ can be nonzero only if $\gamma \ge w_0\lambda \text{.}$ Thus the $M_{\lambda },$ $\lambda \in P^{+},$ are linearly independent. The proof for the cases of $m_{\mu },$ $a_{\mu }$ and $A_{\mu }$ is easier, following directly from (2.5), the fact that $C=\{x\in {ℝ}^n\mid ⟨x,{\alpha }_i^{\vee }⟩>0\text{for all}1\le i\le n\}$ is a fundamental chamber for the action of $W,$ and that if $\mu \in P^{+}\setminus P^{++}$ then $⟨\mu ,{\alpha }_i^{\vee }⟩=0$ and $a_{\mu }=-a_{s_i\mu }=-a_{\mu },$ in which case $a_{\mu }=0$ (similarly for $A_{\mu }\text{).}$ $\square$

For $\lambda \in P$ define the Schur function, or Weyl character, by

$$\begin{array}{cc}{s}_{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }},\text{where}\rho =\frac{1}{2}\sum _{\alpha \in R^{+}}\alpha \text{.}& \text{(2.6)}\end{array}$$

The straightening law for $a_{\mu }$ in (2.5) implies the following straightening law for the Schur functions. If $\mu \in P$ and $w\in W$ then, by (2.5) and the definition of $s_{\mu },$

$$\begin{array}{cc}{(-1)}^{\ell \left(w\right)}{s}_{\mu }=\frac{{(-1)}^{\ell \left(w\right)}{a}_{\mu +\rho }}{a_{\rho }}=\frac{a_{w(\mu +\rho )-\rho +\rho }}{a_{\rho }}=s_{w\circ \mu },& \text{(2.7)}\end{array}$$

where $w\circ \mu =w\left(\mu +\rho \right)-\rho \text{.}$

The dot action of the Weyl group $W$ on ${𝔥}_{ℝ}^{*}$ which is appearing here, $w\circ \mu =t_{-\rho }w{t}_{\rho }\mu =\left(t_{\rho }^{-1}\right)w{t}_{\rho }\mu ,$ is ordinary action of $W$ on ${𝔥}_{ℝ}^{*}$ except with the "center" shirted to $-\rho \text{.}$ For the root system of type $C_2,$ in Example 1.1, the picture is

$$\begin{array}{cc} \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ C \] \[ s_{1}C \] \[ s_{2}C \] \[ s_1{s}_{2}C \] \[ s_2{s}_{1}C \] \[ s_1{s}_2{s}_{1}C \] \[ s_2{s}_1{s}_{2}C \] \[ s_1{s}_2{s}_1{s}_{2}C \] \[ \rho \] \[ s_1\rho \] \[ s_2\rho \]& \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ C \] \[ 0 \] \[ s_1\circ 0 \] \[ s_2\circ 0 \] \[ -\rho \]\\ \text{the orbit}W\rho & \text{the orbit}W\circ 0\end{array}$$

The following proposition shows that the Weyl characters $s_{\lambda }$ are elements of $𝕂{\left[P\right]}^W\text{.}$ The equaility in part (a) is the Weyl denominator formula, a generalization of the factorization of the Vandermonde determinant $\text{det}\left(x_i^{n-j}\right)=\prod _{1\le i,j\le n}(x_i-x_j)\text{.}$ In the remainder of this section we shall abuse language and use the term "vector space" in place of "free $𝕂=\mathbb{Z}[q,q^{-1}]$ module".

Let $P^{+},$ $P^{++},$ $𝕂{\left[P\right]}^W$ and $𝒜$ be as in (1.7) and (2.4) and let $\rho$ be as in (1.8).

1. If $f\in 𝒜$ then $f$ is divisible by $a_{\rho }$ and $$a_{\rho }=x^{\rho }\prod _{\alpha \in R^{+}}(1-x^{-\alpha })$$
2. The set $\{s_{\lambda }\mid \lambda \in P^{+}\}$ is a basis of $𝕂{\left[P\right]}^W\text{.}$
3. The maps $$\begin{array}{ccc}{P}^{+}& ⟶& P^{++}\\ \lambda & ⟼& \lambda +\rho \end{array}\text{and}\begin{array}{cccc}\Phi :& 𝕂{\left[P\right]}^W& \to & 𝒜\\ & f& ⟼& a_{\rho }f\\ & s_{\lambda }& ⟼& a_{\lambda +\rho }\end{array}$$ are a bijection and a vector space isomorphism, respectively.

		Proof.
	Since $s_i$ takes ${\alpha }_i$ to $-{\alpha }_i$ and permutes the other elements of $R^{+},$ $$\rho -⟨\rho ,{\alpha }_i^{\vee }⟩{\alpha }_i=s_i\rho =\rho -{\alpha }_i,$$ and so $$⟨\rho ,{\alpha }_i^{\vee }⟩=1,\text{for all}1\le i\le n\text{.}$$ Thus the map $P^{+}\to P^{++}$ given by $\lambda \mapsto \lambda +\rho$ is well defined and it is a bijection since it is invertible. Let $d=x^{\rho }\prod _{\alpha \in R^{+}}(1-x^{-\alpha })=\prod _{\alpha \in R^{+}}(x^{\alpha /2}-x^{-\alpha /2})\text{.}$ Since $s_i$ takes ${\alpha }_i$ to $-{\alpha }_i$ and permutes the other elements of $R^{+},$ $s_{i}d=-d$ for all $1\le i\le n$ and so $wd={(-1)}^{\ell \left(w\right)}d$ for all $w\in W\text{.}$ Thus $d$ is an element of $𝒜\text{.}$ If $\alpha \in R^{+}$ and $f={\displaystyle \sum _{\mu \in P}}{c}_{\mu }{x}^{\mu }\in 𝒜$ then $$\sum _{\mu \in P}{c}_{\mu }{x}^{\mu }=f=-s_{\alpha }f=\sum _{\mu \in P}-c_{\mu }{x}^{s_{\alpha }\mu },$$ and $$f=\sum _{⟨\mu ,{\alpha }^{\vee }⟩\ge 0}{c}_{\mu }(x^{\mu }-x^{s_{\alpha }\mu })\text{.}$$ Since $(1-x^{-⟨\mu ,{\alpha }^{\vee }⟩\alpha })$ is divisible by $(1-x^{-\alpha })$ it follows that $x^{\mu }-x^{s_i\mu }=x^{\mu }(1-x^{-⟨\mu ,{\alpha }^{\vee }⟩\alpha })$ is divisible by $(1-x^{-\alpha })$ and thus that $f$ is divisible by $(1-x^{-\alpha })$ for all $\alpha \in R^{+}\text{.}$ Since the elements $(1-x^{-\alpha })$ are relatively prime in the Laurent polynomial ring $𝕂\left[P\right]$ and $x^{\rho }$ is a unit in $𝕂\left[P\right],$ $f$ is divisible by $d\text{.}$ Since both $f$ and $d$ are in $𝒜,$ the quotient $f/d$ is an element of $𝕂{\left[P\right]}^W\text{.}$ The monomial $x^{\rho }$ appears in $a_{\rho }$ with the coefficient 1 and it is the unique term $x^{\mu }$ in $a_{\rho }$ with $\mu \in P^{+}\text{.}$ Since $d$ has highest term $x^{\rho }$ with coefficient 1 and $a_{\rho }$ is divisible by $d$ and it follows that $a_{\rho }/d=1\text{.}$ Thus $a_{\rho }=d,$ the inverse of the map $\Phi$ in (c) is well defined, and $\Phi$ is an isomorphism. Since $\{a_{\lambda +\rho }\mid \lambda \in P^{+}\}$ is a basis of $𝒜$ and the map $\Phi$ is an isomorphism it follows that $\{s_{\lambda }\mid \lambda \in P^{+}\}$ is a $𝕂$-basis of $𝕂{\left[P\right]}^W\text{.}$ $\square$

Acknowledgements

The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977 and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).

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