Poincaré series of the Weyl groups of the elliptic root systems \(A_1^{(1,1)}\), \(A_1^{(1,1)^*}\) and \(A_2^{(1,1)}\).
The Poincaré series \(W(t)\) of a group \(W\) with respect to a generator system is defined by \(W(t)=\sum_{w\in W}t^{l(w)}\), where \(t\) is an indeterminate and \(l(w)\) is the length of a minimal expression of an element \(w\) in \(W\) in terms of the given generator system. If \(W\) is one of the finite or affine Weyl groups, it is known that \[ \sum_{w\in W}t^{l(w)}=\begin{cases}\prod_{i=1}^n\tfrac{1-t^{m_i+1}}{1-t}&\text{(\(W\) finite)}\\ \tfrac1{(1-t)^n}\prod_{i=1}^n\tfrac{1-t^{m_i+1}}{1-t}&\text{(\(W\) affine)}\end{cases} \] where \(n\) is the rank of \(W\) and \(m_1,\dots,m_n\) are the exponents of \(W\). In this paper, the author calculates the Poincaré series \(W(t)\) of the elliptic Weyl groups \(W\) of types \(A_1^{(1,1)}\), \(A_1^{(1,1)^*}\) and \(A_2^{(1,1)}\).
scientific article; zbMATH DE number 1981397
scientific article