Lower bounds for degrees of irreducible Brauer characters of finite general linear groups (Q1970974)

Let \(G=\text{GL}_n(\mathbb{F}_q)\), the general linear group over the finite field with \(q\) elements. To each partition \(\lambda\) of \(n\) there is an irreducible unipotent complex character \(X_\lambda\) of \(G\) whose degree is a polynomial in \(q\) given by Green's hook formula [\textit{J. A. Green}, Trans. Am. Math. Soc. 80, 402-447 (1955; Zbl 0068.25605)]. This polynomial is monic of degree \(b(\lambda')\) where \(\lambda'\) is the transpose of \(\lambda\) and \[ b\bigl[\mu=(m_1\geq m_2\geq\cdots\geq m_h>0)\bigr]=\tfrac{n(n+1)}2-\sum^h_{i=1}im_i. \] From Green's formula one also can deduce that \(X_\lambda(1)\geq q^{b(\lambda')}\). In this paper the authors prove similar lower bounds for the degrees of the irreducible \(p\)-modular Brauer characters of \(G\) when \(p\) is coprime to \(q\).