Semi-infinite Plücker Relations and Weyl Modules

Abstract

The goal of this paper is twofold. First, we write down the semi-infinite Plücker relations, describing the Drinfeld-Plücker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, i.e. the quotient by the ideal generated by the semi-infinite Plücker relations. We establish the isomorphism with the algebra of dual global Weyl modules and derive a new character formula.

Vertex Algebras and Coordinate Rings of Semi-infinite Flags

“…We apply the resulting formula to derive a set of quadratic relations in the algebra W * . We show that in type A these relations are defining and coincide with those given in [FM2].…”

“…Proof. Corollary 3.22 and Proposition 2.9 a), [FM2] state that the ideal of relations of W * is generated by its quadratic part and the generating set of relations can be described as follows: given a non semi-standard two-column Young tableaux T = (I, J) of shape ω i + ω j one produces a relation in the coordinate ring where X I (z) = k≥0 (X I t −k )z k and X I t −k is considered as an element of W * ω l(I) [t −1 ] ≃ W * ω l (I) . Corollary 3.22 and Proposition 2.9 a), [FM2] state that the coefficients of the relations above generate the ideal of relations in W * .…”

“…So we only need to show that the summand V ω j−l +ω i+l shows up at the level l in W ω i +ω j . Theorem 3.21 of [FM2] gives the following formula for the character of W ω i +ω j (here x a = exp(ε a )): We claim that for 1 ≤ l ≤ min(j, r + 1 − i) (6.1) ch V ω j−l +ω i+l = |I|=i,|J|=j k(I,J)=l l(I) a=1…”

“…Let us now explicitly describe the relations for the algebra W * . In [FM2] for any two strictly increasing columns I and J such that length of I is no smaller than the length of J we defined the number k(I, J). The value k(I, J) measures how far the two-column Young tableaux Y (I, J) corresponding to the pair I, J is from being semi-standard.…”

“…It is tempting to conjecture that the same result hold in all types. Our argument is based on [FM2], which treats type A only. For other types it is not even known if the ideal of relations is generated by quadratic part.…”