On the sum of matrices of special linear group over finite field

Abstract

Let $ \mathbb{F}_q $ be a finite field of $ q $ elements. For $ n\in\mathbb{N}^* $ with $ n\ge2, $ let $ M_{n}: = Mat_n(\mathbb{F}_q) $ be the ring of matrices of order $ n $ over $ \mathbb{F}_q, $ $ G_{n, 1}: = Sl_n(\mathbb{F}_q) $ be the special linear group over $ \mathbb{F}_q. $ In this paper, by using the technique of Fourier transformation, we obtain a formula for the number of representations of any element of $ M_{n} $ as the sum of $ k $ matrices in $ G_{n, 1}. $ As a corollary, we give another proof of the number of the third power moment of the classic Kloosterman sum.

Keywords

• ring of matrices
• finite field
• sum of matrices
• fourier transformation
• kloosterman sum