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\title{Canonical $p$-dimensions of algebraic groups and
degrees of basic polynomial invariants}

\author{K.~Zainoulline}


%%%%%%%%%%%%%%%%%%%% DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

\begin{abstract}
In the present notes we provide a new uniform way to compute 
a canonical $p$-dimension of a split algebraic group $G$ for
a torsion prime $p$
using degrees of basic polynomial invariants described by V.~Kac.
As an application, we compute the canonical $p$-dimensions
for all split simple algebraic groups.
\end{abstract}

The notion of a canonical dimension of an algebraic structure
was introduced by Berhuy and Reichstein \cite{BR}.
For a split algebraic group $G$ and its torsion prime $p$ 
the canonical $p$-dimension of $G$ was studied by 
Karpenko and Merkurjev in \cite{Ka} and \cite{KM}.
In particular, this invariant was shown to be related with
the size of the image of the characteristic map 
\begin{equation}
\phi_G: S^*(\hat{T}) \to \CH^*(X),
\end{equation}
where $\hat{T}$ is the character group of a maximal split torus $T$,
$X=G/B$ is the variety of complete flags and $S^*$ stands for the
 symmetric algebra.
Namely, one has the following formula for 
the canonical $p$-dimension of a group $G$
\begin{equation}
\cd_p(G)=\min \{i \mid \overline{\Ch_i}(X)\},
\end{equation}
where $\overline{\Ch_i}(X)$ stands for the image of $\phi_G$ in the Chow
ring $\Ch_i(X)=\CH_i(X;\zz/p\zz)$ with $\zz/p\zz$-coefficients.
Using this remarkable fact together with the explicit description
of the image $R_p=\overline{\Ch}(X)$
Karpenko and Merkurjev computed the canonical $p$-dimensions
for all classical algebraic groups $G$.


The goal of the present notes is to relate
the work by V.~Kac \cite{Kc}
devoted to the study of the $p$-torsion of the Chow ring of an algebraic group $G$
with the canonical $p$-dimensions of $G$.
In particular we provide a different and uniform approach
of computing $cd_p(G)$.

\begin{mainthm} Let $G$ be a split simple algebraic group of rank $n$
and $p$ be a torsion prime. 
Then
$$
\cd_p(G)=N + n - (d_{1,p}+ d_{2,p}+\ldots+d_{n,p}),
$$
where $N$ stands for the number of positive roots of $G$ and 
integers $d_{1,p},\ldots,d_{1,n}$ are 
the degrees of basic polynomial invariants modulo $p$.
\end{mainthm}

\begin{proof} Consider the characteristic map (1) modulo $p$
\begin{equation}
(\phi_G)_p: S^*(\hat{T})\otimes_\zz \zz/p\zz \to \Ch^*(X)
\end{equation}
According to \cite[Theorem 1]{Kc}, 
the kernel of this map $I_p$ is generated by a regular sequence
of $n$ homogeneous polynomials of degrees $d_{1,p},\ldots,d_{n,p}$.

Recall that a Poincare polynomial $P(A,t)$ for a graded module $M^*$ 
over a field $k$ is defined to be
$\sum_i \dim_k M^i \cdot t^i$ (see \cite{St}).
Hence, the Poincare polynomial for the $\zz/p\zz$-module 
$R_p$ is equal to
\begin{equation}
P(R_p,t)= \prod_{i=0}^n \frac{1-t^{d_{i,p}}}{1-t}.
\end{equation}
Indeed, we identify $R_p$ with the quotient of the polynomial ring
in $n$ variables $\zz/p\zz[\omega_1,\ldots,\omega_n]$ modulo the ideal $I_p$
(here $\omega_i$ are the fundamental weights).
The formula (4) then follows immediately by \cite[Cor.~3.3]{St}.

According to (2) the canonical $p$-dimension is equal to the difference
$\dim(X)- \deg P(R_p,t)$.
\end{proof}

\begin{cor} We obtain the following values for the canonical $p$-dimensions
of groups of types $F_4$, $E_6$, $E_7$ and $E_8$ (in the list below
$G^{sc}$ and $G^{ad}$ denote the simply-connected and adjoint forms of $G$).

\begin{tabular}{ccc}
$\cd_2 F_4 = 3$, & $\cd_3 F_4 = 8$ & \\
$\cd_2 E_6 = 3$, & $\cd_3 E_6^{sc} = 8$, & $\cd_3 E_6^{ad} = 16$ \\
$\cd_2 E_7^{sc} = 17$, & $\cd_2 E_7^{ad} = 18$ & $\cd_3 E_7 = 8$ \\
$\cd_2 E_8 = 60$, & $\cd_3 E_8= 28$ & $\cd_5 E_8= 24$ 
\end{tabular}

\end{cor}

\begin{proof} Follows from the list of degrees of basic polynomial invariants
provided in \cite[Table 2]{Kc}.
\end{proof}

\begin{cor}
There is the following relation between
the canonical $p$-dimension of $G$ and the $p$-torsion part of the Chow group of $G$
$$
\cd_p(G)= \max \{i \mid \Ch^i(G)\neq 0 \}
$$
\end{cor}

\begin{proof} It is known that $\Ch(G)=\Ch(X)/J_p$,
where $J_p$ is the ideal generated by the non-constant part of $R_p$.
Since $\Ch(X)$ is a free $R_p$-module, we obtain
$P(\Ch(X)/J_p,t)=P(\Ch(X),t)/P(R_p,t)$.
\end{proof}

\begin{cor}
Let $d_1,d_2,\ldots,d_n$ be the degrees of ``usual'' basic polynomial
invariants of $G$, i.e., with $\mathbb{Q}$-coefficients. Then the canonical
$p$-dimension of $G$ is equal to
$$
\cd_p(G) = \sum_{i=0}^n d_i'\cdot (p^{k_i}-1),
$$
where the integers $d_i'$ and $p^{k_i}$ are the factors of the decompositions
$d_i=d_i'\cdot p^{k_i}$, $p \nmid d_i'$. Observe that the degrees $d_i$ for
which $k_i>0$ form the set of $p$-exceptional degrees introduced in \cite{Kc}.
\end{cor}

\begin{proof} Follows by the isomorphism of \cite[Theorem 3,(ii)]{Kc}.
\end{proof}

{\it Acknowledgements.} 
I am grateful to Burt Totaro for mentioning to me
the paper \cite{Kc}.

\bibliographystyle{chicago}
\begin{thebibliography}{99}



\bibitem{BR} Berhuy, G., Reichstein, Z. On the notion of canonical dimension
for algebraic groups. Advances in Math., in press, doi: 10.1016/j.aim.2004.12.004.

\bibitem{Kc} Kac, V. Torsion in cohomology of compact Lie groups and
Chow rings of reductive algebraic groups. Invent. Math. 80 (1985), 69--79.

\bibitem{Ka} Karpenko, N. Canonical dimension of orthogonal group. Transform. Groups 10 (2005), no.~2, 211--215.

\bibitem{KM} Karpenko, N., Merkurjev, A. Canonical $p$-dimension of algebraic groups.
Advances in Math., in press, doi: 10.1016/j.aim.2005.07.013.

\bibitem{St} Stanley, R. Hilbert functions of graded algebras. Advances in Math. 28 (1978), 57--83.

\end{thebibliography}

\end{document}
