## Symmetric and alternating functions and their $q$-analogues

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

$102=10 and Ti10= q10, for all1≤i≤n, ε02=ε0 and Tiε0= (-q-1) ε0, for all1≤i≤n.$

In terms of the basis $\left\{{T}_{w}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\in W\right\}$ of $H$ these elements have the explicit formulae

$10= 1W0(q2) ∑w∈W qℓ(w)Tw, andε0= 1W0(q-2) ∑w∈W (-q)-ℓ(w) Tw, (2.1)$

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

$1=1∣W∣ ∑w∈Wwand ε=1∣W∣ ∑w∈W (-1)ℓ(w)w, (2.2)$

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 $𝕂=ℤ\left[q,{q}^{-1}\right]$-modules) of symmetric functions and alternating functions,

$𝕂[P]W = { f∈𝕂[P]∣ wf=ffor allw∈W } =1𝕂[P], 𝒜 = { f∈𝕂[P]∣ wf=(-1)ℓ(w) ffor allw∈W } =ε𝕂[P], (2.3)$

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

$Wμ= {wμ∣w∈W} andWμ= {w∈W∣wμ=μ} .$

Then define

$mμ= ∑γ∈Wμ xγ= ∣W∣ ∣Wμ∣ 1xμ, aμ=∑w∈W (-1)ℓ(w) wxμ=∣W∣ε xμ, Mμ=10 xμ10, Aμ=ε0xμ 10. (2.4)$

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

$msiμ=mμ, asiμ=-aμ, andAsiμ= -Aμ.$

Letting $t={q}^{-2},$ ${M}_{\mu }={M}_{{s}_{i}\mu }$ if $d=0,$ and if $d>0$ then

$Msiμ = tMμ + ( ∑ j=1 ⌊d/2-1⌋ (t2-1) tj-1 Mμ-jαi ) + { (t-1)td/2-1 Mμ-(d/2)αi , ifdis even, 0 , ifdis odd.$

 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 ${x}^{\gamma },$ $\gamma \in P,$ $Aμ+Asiμ = ε0 (xμ+xsiμ) 10=q-1ε0 (xμ+xsiμ) Ti10 = q-1ε0Ti (xμ+xsiμ) 10=-q-2 ε0 (xμ+xsiμ) 10 = -q-2 (Aμ+Asiμ) .$ 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 $10 (xsiμ-xμ) 10=q-1 (q-q-1)10 ( xsiμ-xμ 1-x-αi ) 10.$ Subtracting the same relation with $\mu$ replaced by $\mu -{\alpha }_{i}$ gives $10 (xsiμ-xμ) 10-10 ( xsiμ+αi- xμ-αi ) 10 = (1-q-2)10 ( xsiμ-xμ- xsiμ+αi+ xμ-αi 1-x-αi ) 10 = (1-q-2)10 ( -xsiμ+αi -xμ ) 10.$ So $10xsiμ10 =q-210xμ 10-10 xμ-αi10+ q-210 xsiμ+αi10.$ Inductively applying this relation yields the result. The first cases are $Msiμ= { Mμ , if⟨μ,αi∨⟩ =0, q-2Mμ , if⟨μ,αi∨⟩ =1, q-2Mμ+ (q-2-1) Mμ-αi , if⟨μ,αi∨⟩ =2, q-2Mμ+ (q-4-1) Mμ-αi , if⟨μ,αi∨⟩ =3, q-2Mμ+ (q-4-1) Mμ-αi +q-2 (q-2-1) Mμ-2αi , if⟨μ,αi∨⟩ =4.$ $\square$

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

$mwμ=mμ, awμ= (-1)ℓ(w) aμ,and Awμ (-1)ℓ(w) Aμ. (2.5)$

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

$𝕂[P]W has basis {mλ∣λ∈P+}, 10H∼10 has basis {Mλ∣λ∈P+}, 𝒜 has basis {aμ∣μ∈P++}, ε0H∼10 has basis {Aμ∣μ∈P++}.$

 Proof. Since $\left\{{x}^{\mu }{T}_{w}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\mu \in P,w\in W\right\}$ 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, $\gamma \in {P}^{+}=P\cap \stackrel{‾}{C},$ where $\stackrel{‾}{C}=\left\{x\in {ℝ}^{n}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}⟨x,{\alpha }_{i}^{\vee }⟩\ge 0\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}1\le i\le n\right\}\text{.}$ If $\lambda \in {P}^{},$ using the fourth relation in (1.21), $Mλ = 10xλ10= 1W0(q2) ∑w∈W qℓ(w)Tw xλ10= 1W0(q2) ∑γ,v,w qℓ(w) dv,γxγ Tv10 = 1W0(q2) ∑γ,v,w qℓ(w) dv,γxγ qℓ(v)10= 1W0(q2) ∑γdγxγ10,$ where ${d}_{v,\gamma }$ and ${d}_{\gamma }$ are some polynomials in $ℤ\left[q,\left(q-{q}^{-1}\right)\right]$ 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 $C=\left\{x\in {ℝ}^{n}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}⟨x,{\alpha }_{i}^{\vee }⟩>0\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}1\le i\le n\right\}$ 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

$sλ=aλ+ρaρ, whereρ=12 ∑α∈R+α. (2.6)$

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 ${s}_{\mu },$

$(-1)ℓ(w) sμ= (-1)ℓ(w) aμ+ρ aρ = aw(μ+ρ)-ρ+ρ aρ =sw∘μ, (2.7)$

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

$Hα1+α2 Hα1 Hα2 Hα1+2α2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ρ s1ρ s2ρ Hα1+α2 Hα1 Hα2 Hα1+2α2 C 0 s1∘0 s2∘0 -ρ the orbitWρ the orbitW∘0$

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 $\text{det}\phantom{\rule{0.2em}{0ex}}\left({x}_{i}^{n-j}\right)=\prod _{1\le i,j\le n}\left({x}_{i}-{x}_{j}\right)\text{.}$ In the remainder of this section we shall abuse language and use the term "vector space" in place of "free $𝕂=ℤ\left[q,{q}^{-1}\right]$ 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 $aρ=xρ ∏α∈R+ (1-x-α)$
2. The set $\left\{{s}_{\lambda }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\lambda \in {P}^{+}\right\}$ is a basis of $𝕂{\left[P\right]}^{W}\text{.}$
3. The maps $P+⟶P++ λ⟼λ+ρ and Φ:𝕂[P]W →𝒜 f⟼aρf sλ⟼aλ+ρ$ 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 ${R}^{+},$ $ρ- ⟨ρ,αi∨⟩ αi=siρ=ρ -αi,$ and so $⟨ρ,αi∨⟩ =1,for all1≤ i≤n.$ Thus the map ${P}^{+}\to {P}^{++}$ given by $\lambda ↦\lambda +\rho$ is well defined and it is a bijection since it is invertible. Let $d={x}^{\rho }\prod _{\alpha \in {R}^{+}}\left(1-{x}^{-\alpha }\right)=\prod _{\alpha \in {R}^{+}}\left({x}^{\alpha /2}-{x}^{-\alpha /2}\right)\text{.}$ 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 $wd={\left(-1\right)}^{\ell \left(w\right)}d$ for all $w\in W\text{.}$ Thus $d$ is an element of $𝒜\text{.}$ If $\alpha \in {R}^{+}$ and $f=\sum _{\mu \in P}{c}_{\mu }{x}^{\mu }\in 𝒜$ then $∑μ∈Pcμxμ =f=-sαf= ∑μ∈P-cμ xsαμ,$ and $f=∑⟨μ,α∨⟩≥0 cμ (xμ-xsαμ).$ Since $\left(1-{x}^{-⟨\mu ,{\alpha }^{\vee }⟩\alpha }\right)$ is divisible by $\left(1-{x}^{-\alpha }\right)$ it follows that ${x}^{\mu }-{x}^{{s}_{i}\mu }={x}^{\mu }\left(1-{x}^{-⟨\mu ,{\alpha }^{\vee }⟩\alpha }\right)$ is divisible by $\left(1-{x}^{-\alpha }\right)$ and thus that $f$ is divisible by $\left(1-{x}^{-\alpha }\right)$ for all $\alpha \in {R}^{+}\text{.}$ Since the elements $\left(1-{x}^{-\alpha }\right)$ 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 $\left\{{a}_{\lambda +\rho }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\lambda \in {P}^{+}\right\}$ is a basis of $𝒜$ and the map $\Phi$ is an isomorphism it follows that $\left\{{s}_{\lambda }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\lambda \in {P}^{+}\right\}$ 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).