A Gelfand–Tsetlin type basis for the algebra 𝔤₂

Abstract

<p> The paper presents a construction of finite-dimensional irreducible representations of the Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {g}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The representation space is constructed as the space of solutions to a certain system of partial differential equations of hypergeometric type, which is closely related to the Gelfand–Kapranov–Zelevinsky systems. This relationship allows for the construction of a basis in the representation. An orthogonalization of the constructed basis with respect to an invariant scalar product turns out to be a Gelfand–Tsetlin-type basis for the chain of subalgebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g 2 superset-of German s German l Subscript 3"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo> ⊃ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {g}_2 \supset \mathfrak {sl}_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . </p>