Crystal bases for the quantum superalgebra 𝑈_{𝑞}(𝔤𝔩(𝔪,𝔫))

Abstract

A crystal base theory is introduced for the quantized enveloping algebra of the general linear Lie superalgebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German l left-parenthesis m comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</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">\mathfrak {gl}(m,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and an explicit realization of the crystal base is given in terms of semistandard tableaux.