Part I

A conjecture on Weyl's dimension polynomial

Last update: 2 April 2014

**2.1.** The polynomial referred to in the title of Part I is the following function on $E\text{:}$
$$D\left(\lambda \right)=\frac{\prod _{\alpha \in {\Delta}_{+}}\u27e8\lambda ,{\alpha}^{\vee}\u27e9}{\prod _{\alpha \in {\Delta}_{+}}\u27e8\delta ,{\alpha}^{\vee}\u27e9}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}\lambda \in E,$$
where $\delta =\sum _{i}{\lambda}_{i}=\frac{1}{2}\sum _{\alpha \in {\Delta}_{+}}\alpha ,$
as usual. Because of homogeneity one could replace ${\alpha}^{\vee}$ by $\alpha $ both in the numerator and
the denominator. But the form given here emphasizes the fact that if we consider $D$ as a polynomial in the coefficients
${x}_{1},{x}_{2},\dots ,{x}_{l}$
in the expansion of $\lambda $ in the fundamental weights, $\lambda =\sum _{i}{x}_{i}{\lambda}_{i},$
then the denominator is an integer and the numerator is a product of linear forms in ${x}_{1},{x}_{2},\dots ,{x}_{l}$
(called root-forms) having non-negative integer coefficients; these forms $\u27e8\lambda ,{\alpha}^{\vee}\u27e9$
can be easily written down in any given case once the Dynkin diagram of $\Delta $ is given. The value of
$D\left(\lambda \right)$ is therefore $\geqq 0$ for
$\lambda \in {X}^{+},$ and that of
$D(\lambda +\delta )$ is strictly positive; in fact, the latter number is
an integer, being the dimension of the "irreducible ${\U0001d524}_{\u2102}\text{-module}$ with highest weight
$\lambda \text{",}$ where ${\U0001d524}_{\u2102}$ is the simple Lie algebra over
$\u2102$ possessing $\Delta $ as its root system. (It should be noted that when the "coordinates"
${x}_{1},{x}_{2},\dots ,{x}_{l}$
of $\lambda =\sum _{i}{x}_{i}{\lambda}_{i}$
are non-negative integers, i.e. $\lambda \in {X}^{+},$ those of
$\lambda +\delta $ are strictly positive integers, and conversely; that justifies the parenthetical statement in the opening
sentence of the Introduction.)

The polynomial $D\left(\lambda \right)$ is determined by $\Delta $ and the choice of positive roots therein. But a different choice of ${\Delta}_{+}$ (equivalently, of the fundamental roots) affects $D\left(\lambda \right)$ only by a multiplicative factor of $-1\text{.}$ This is because all choices of ${\Delta}_{+}$ are conjugate under $W,$ and $D$ is skew (or alternating) with respect to $W$ in the sense that $D\left({\lambda}^{\sigma}\right)=\text{det}\hspace{0.17em}\sigma \xb7D\left(\lambda \right)$ for $\sigma \in W\text{.}$ As a matter of fact, it is known that every skew-invariant polynomial on $E$ is of the form $D\left(\lambda \right)\xb7P\left(\lambda \right)$ where. $P$ is $W\text{-invariant.}$ There is another property that distinguishes $D,$ viz. that up to scalar multiples it is the unique $W\text{-harmonic}$ homogeneous polynomial of maximal degree. A polynomial on $E$ is called $W\text{-harmonic}$ if it is annihilated by all $W\text{-invariant}$ homogeneous differential operators of positive degree. (We consider only operators with constant coefficients. The algebra of such operators is canonically isomorphic to the symmetric algebra $S\left(E\right)$ of $E\text{;}$ for $u\in S\left(E\right)$ one denotes the corresponding differential operator by ${\partial}_{u}\text{.}$ The action of ${\partial}_{u}$ on the algebra of polynomial functions on $E,$ which is simply the algebra $S\left({E}^{*}\right),$ is obtained by requiring that for $p\in S\left({E}^{*}\right)$ the value of ${\partial}_{u}p$ at $x\in E$ be equal to the inner-product $\u27e8u,{p}^{x}\u27e9$ where ${p}^{x}$ is the translated polynomial ${p}^{x}\left(\lambda \right)=p(\lambda +x)$ and $\u27e8,\u27e9$ is the canonical pairing between $S\left(E\right)$ and $S\left({E}^{*}\right)$ extending that between $E$ and ${E}^{*}\text{.}$ When we consider a polynomial function on $E$ as a formal polynomial in the coordinates ${x}_{1},{x}_{2},\dots ,{x}_{l}$ of $\lambda =\sum _{i}{x}_{i}{\lambda}_{i},$ we can identify the first-order homogeneous differential operator ${\partial}_{{\lambda}_{i}}$ in the above notation with the formal partial derivation $\frac{\partial}{\partial {x}_{i}}$ in the standard sense.)

The fundamental fact about $W\text{-harmonic}$ polynomials (cf. Steinberg [Ste1964]) is that the space of all such polynomials has dimension $\left|W\right|$ and is spanned by $D$ and all its (partial) derivatives. This generalizes, in fact, to the situation when $W$ is an arbitrary finite pseudoreflection group (in the accepted Bourbaki usage), which includes the noncrystallographic $\text{(}=$ non-Weyl) finite Coxeter groups; only the definition of $D$ has to be slightly modified to take into account the orders of the generating pseudoreflections which can be greater than 2.

**2.2.** We can now state our conjecture on $D$ in terms of the affine transformations
${w}_{\sigma}\in \stackrel{\sim}{W}$ (for
$\sigma \in W\text{)}$ introduced in § 1; we remark that though $D$ depends only on
$W$ this conjecture depends on the extension $\stackrel{\sim}{W}$ (as isomorphic Weyl groups can give rise to
distinct extensions), and does not make sense for even the noncrystallographic finite Coxeter groups.

*Conjecture * I.
*
There exist unique scalars ${b}_{\sigma},$
$\sigma \in W,$ such that
$$\begin{array}{cc}\sum _{\sigma \in W}{b}_{\sigma}\xb7D\left({\lambda}^{{w}_{\sigma}}\right)=1\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{1em}{0ex}}\lambda \in E\text{;}& \text{(2.1)}\end{array}$$
these "weightings" ${b}_{\sigma}$ are necessarily integers.
*
(See ${\text{Footnote}}^{\text{2}}\text{.)}$

Let us write ${D}_{\sigma}\left(\lambda \right)=D\left({\lambda}^{{w}_{\sigma}}\right)=\text{det}\hspace{0.17em}\sigma \xb7D(\lambda +{\left({\delta}_{\sigma}\right)}^{\sigma -1}),$ and note that ${D}_{\sigma}$ is a harmonic polynomial for all $\sigma \text{.}$ In fact for arbitrary $\mu \in E$ the translate ${H}^{\mu}$ of a harmonic polynomial $H$ is again harmonic, as is clear from Taylor's theorem and the fact that the derivatives of $H$ are harmonic. It follows that the first half of the conjecture simply asserts the linear independence of the set $\left\{{D}_{\sigma}\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W\}$ of inhomogeneous polynomials. Since ${\left({\delta}_{\sigma}\right)}^{\sigma -1}\in X,$ all the ${D}_{\sigma}$ belong to the $\mathbb{Q}\text{-span}$ of $D$ and its successive derivatives with respect to ${\partial}_{\lambda}$ for $\lambda \in X\text{;}$ since this $\mathbb{Q}\text{-span}$ is a $\mathbb{Q}\text{-form}$ of the space of harmonic polynomials, it is clear that all ${b}_{\sigma}$ are rational numbers. Thus the crux of the conjecture lies in the assertion ${b}_{\sigma}\in \mathbb{Z}\text{.}$ However, it should be emphasized that our main interest is not in the existence and uniqueness of ${b}_{\sigma}\text{'s,}$ but in finding an explicit method of computing their values. Unfortunately, even the examination of several low rank cases in the next section sheds little light on this problem.

**2.3.** Our "inhomogeneous weighted mean-value conjecture" above has a known
homogeneous analogue which asserts
$$\begin{array}{cc}{\left|W\right|}^{-1}\sum _{\sigma \in W}D(\lambda +{\mu}^{\sigma})=D\left(\lambda \right)\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}\lambda ,\mu \in E\text{.}& \text{(2.2)}\end{array}$$
This is in fact true for an arbitrary finite linear group $W$ with $D$ an arbitrary
$W\text{-harmonic}$ polynomial, and is quite easy to prove: Since the translation operator
${T}_{\mu}$ sending $D\left(\lambda \right)$ to
${D}^{\mu}\left(\lambda \right)=D(\lambda +\mu )$
is a differential operator (Taylor's theorem), with constant term in it equal to 1 (corresponding to the fact that the highest degree homogeneous term in the expansion of
${D}^{\mu}$ is $D$ itself), the $W\text{-invariant}$ operator
$$\left(\sum _{\sigma \in W}{T}_{{\mu}^{\sigma}}\right)-\left|W\right|$$
has zero constant term, and hence annihilates every harmonic polynomial, giving (2.2). This argument is due to Steinberg.

It is conceivable that an appropriate generalization of (2.1) would render it easier to prove. Venturing in one possible direction of generalization, we may ask if the following is true: $$\begin{array}{cc}\sum _{\sigma \in W}{b}_{\sigma}\xb7D(\lambda +{\mu}^{{w}_{\sigma}})\phantom{\rule{1em}{0ex}}\text{is independent of}\phantom{\rule{1em}{0ex}}\mu \in E,& \text{(2.3)}\end{array}$$ where ${b}_{\sigma}$ are as in the conjecture. We have verified this only for the case of type ${A}_{2},$ apart from the trivial case of type ${A}_{1}\text{.}$

**2.4.** Note that there is exactly one value of $\sigma \in W,$ viz. $\sigma ={\sigma}_{0},$
such that ${D}_{\sigma}\left(\lambda \right)=D\left({\lambda}^{{w}_{\sigma}}\right)$
has a nonzero constant term, i.e. does not vanish for $\lambda =0\text{;}$ this is clear from the fact that
${D}_{\sigma}\left(0\right)=D\left({\delta}_{\sigma}\right)\ne 0$
if and only if ${\delta}_{\sigma}$ is $W\text{-regular.}$ Therefore, for
$\lambda =0$ (2.1) specializes to $\sum _{\sigma \in W}{b}_{\sigma}\xb7D\left({\delta}_{\sigma}\right)=1,$
which gives ${b}_{{\sigma}_{0}}=1\text{.}$

Let us also observe that the weighting function $\sigma \to {b}_{\sigma}$ on $W$ is constant on each right-coset with respect to the subgroup ${\Omega}_{0}$ introduced in § 1, i.e. ${b}_{\gamma \sigma}={b}_{\sigma}$ for $y\in {\Omega}_{0}$ and $\sigma \in W\text{.}$ This is clear from the fact that for $\lambda $ in the interior of ${E}^{*}$ one has $$\sum _{\sigma}{b}_{\gamma \sigma}{D}_{\sigma}\left(\lambda \right)=\sum _{\sigma}{b}_{\sigma}{D}_{{\gamma}^{-1}\sigma}\left(\lambda \right)=\sum _{\sigma}{b}_{\sigma}D\left({\lambda}^{\left({\gamma}^{-1}\sigma \right)}\right)=\sum _{\sigma}{b}_{\sigma}D\left({\left({\lambda}^{\left({\gamma}^{-1}\right)}\right)}^{\left(\sigma \right)}\right)=1\text{.}$$ Thus the weighting function is determined by its restriction to the set ${W}^{\u266f}$ of distinguished coset representatives for ${\Omega}_{0}\backslash W,$ and (2.1) becomes $$\sum _{\sigma \in {W}^{\u266f}}{b}_{\sigma}\left(\sum _{j\in {J}_{0}}D\left({\lambda}^{{\omega}_{j}{w}_{\sigma}}\right)\right)=1\text{.}$$

${}^{2}$ *(Added in Proof.)* In the original (preprint) version of this paper it was hoped that
${b}_{\sigma}\text{'s}$ would be necessarily non-negative (as is the case for rank 2 and for type
${A}_{3}$ cf. § 3). But shortly afterwards (in Autumn 1972) S. G. Hulsurkar found methods for hand-computation of the
numbers ${b}_{\sigma}$ (more efficient than our brutal technique in § 3), and carrying it through for types
${B}_{3}$ and ${C}_{3}$ discovered that in both these cases some of the
${b}_{\sigma}\text{'s}$ are negative. Later, refining on his methods, he proved this conjecture in a
form stronger than stated (and required) by us, by showing that the set $\left\{{D}_{\sigma}\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W\}$
is a $\mathbb{Z}\text{-basis}$ of the "lattice" of those harmonic polynomials that take integer values on the weight-lattice
$X\text{.}$ The mystery shrouding the values of ${b}_{\sigma}$ has also disappeared,
as it turns out that ${b}_{\sigma}=\sum _{\tau}{b}_{\sigma ,\tau}$
where $\left[{b}_{\sigma ,\tau}\right]$ is the integer matrix inverse to
the $\left|W\right|\times \left|W\right|$ matrix
$\left[D(-{\delta}_{\sigma}^{{\sigma}^{-1}\tau}+{\delta}_{\tau})\right]\text{;}$
the cruical step in Hulsurkar's work rests in the discovery that the latter matrix is unimodular.

In a forthcoming paper I shall show that the integers ${b}_{\sigma}$ arise naturally as the degrees of certain (unique) virtual
$T\text{-modules}$ given by ${\beta}_{\sigma}\in \mathbb{Z}\xb7{e}^{X}$
(for notations see § 4) such that
$${e}^{\lambda}=\sum _{\sigma}\text{ch}\hspace{0.17em}({\lambda}^{\sigma}+{\delta}_{\sigma})\xb7{\beta}_{\sigma}\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{1em}{0ex}}\lambda \in X,\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}\text{ch}\hspace{0.17em}\mu =\frac{\sum \text{det}\hspace{0.17em}\tau \xb7{e}^{{\mu}^{\tau}}}{\sum \text{det}\hspace{0.17em}\tau \xb7{e}^{{\delta}^{\tau}}}$$
This formula refines Conjecture I on one hand (taking "trace" on both sides one gets (2.1)), and implies on the other hand the fact the ring
$\mathbb{Z}\xb7{e}^{X}$ is free as a module over its subring of
$W\text{-invariants;}$ it seems that this fundamental fact of "semisimple Lie Theory" remained unnoticed until topologists
discovered it (it was proved by Harsh Pittie in 1970 who attributes it as a conjecture of Luke Hodgkin). *See also Footnote*
${}^{6}\text{.}$ In subsequent paper(s) I hope to show that these virtual
$T\text{-modules}$ ${\beta}_{\sigma}$ enter even the complex representation theory
of finite Chevalley groups in a fundamental manner.

This is an excerpt of the paper *The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras* by Daya-Nand Verma.
It appeared in Lie Groups and their Representations, *ed. I.M.Gelfand*, Halsted, New York, pp. 653–705, (1975).