Frobenius extensions and the exotic nilCoxeter algebra for 𝐺(𝑚,𝑚,3)

Abstract

<p> In a previous paper of the first author, the type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript n minus 1"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">A_{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> affine Cartan matrix was <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -deformed to produce a deformation of the reflection representation of the affine Weyl group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript a f f"> <mml:semantics> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mi>f</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">W_{aff}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This deformation plays a role in the quantum geometric Satake equivalence. In this paper we introduce the study of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -deformed divided difference operators. </p> <p> When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is specialized to a primitive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 m"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -th root of unity, this reflection representation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript a f f"> <mml:semantics> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mi>f</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">W_{aff}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> factors through a quotient, the complex reflection group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis m comma m comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(m,m,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The divided difference operators now generate a finite-dimensional algebra we call the exotic nilCoxeter algebra. This algebra is new and has surprising features. In addition to the usual braid relations, we prove a new relation called the roundabout relation. </p> <p> A classic result of Demazure, for Weyl groups, states that the polynomial ring of the reflection representation is a Frobenius extension over its subring of invariant polynomials, and describes how the Frobenius trace can be constructed within the nilCoxeter algebra. We study the analogous Frobenius extension for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis m comma m comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(m,m,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and identify the Frobenius trace within the exotic nilCoxeter algebra for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis m comma m comma 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(m,m,3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . </p>