On some Airy sheaves of Laurent type

Abstract We study certain one-parameter families of exponential sums of Airy–Laurent type. Their general theory was developed in Katz and Tiep (Airy sheaves of Laurent type: an introduction, https://web.math.princeton.edu/~nmk/kt31_11sept.pdf ). In the present paper, we make use of that general theory to compute monodromy groups in some particularly simple families (in the sense of “simple to remember"), realizing Weyl groups of type $$E_6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> and $$E_8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>8</mml:mn> </mml:msub> </mml:math> .