# Frobenius manifolds and a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$

## TL;DR

This paper introduces a new class of extended affine Weyl groups, establishes their invariant theory, and constructs Frobenius manifold structures and superpotentials on their orbit spaces, advancing the understanding of algebraic and geometric structures related to Weyl groups.

## Contribution

It defines a novel class of extended affine Weyl groups and develops Frobenius manifold structures and Landau-Ginzburg superpotentials for their orbit spaces, extending previous theories.

## Key findings

- New class of extended affine Weyl groups $ ilde{W}^{(k,k+1)}(A_l)$ introduced.
- Analogues of Chevalley-type theorems established for these groups.
- Frobenius manifold structures and superpotentials constructed on orbit spaces.

## Abstract

We present a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$ for $1\leq k <l$ and obtain an analogue of Chevalley-type theorem for their invariants. We further show the existence of Frobenius manifold structures on the orbit spaces of $\widetilde{W}^{(k,k+1)}(A_l)$ and also construct Landau--Ginzburg superpotentials for these Frobenius manifold structures.