A Straightening Algorithm for Row-Convex Tableaux

Abstract

We produce a new basis for the Schur and Weyl modules associated to a row-convex shape D. The basis is indexed by a new class of ''straight'' tableaux which we introduce by weakening the usual requirements for standard tableaux. Spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis. For skew shapes, this algorithm specializes to the classical straightening law. The new straight basis is used to produce bases for flagged Schur and Weyl modules, to provide Groebner and SAGBI bases for the homogeneous coordinate rings of some configuration varieties, and to produce a flagged branching rule for row-convex representations. Systematic use of supersymmetric letterplace techniques enables the representation theoretic results to be applied to representations of the general linear Lie superalgebra as well as to the general linear group. ᮊ

Row Convex Tableaux and Bott–samelson varieties

“…We now discuss an analogous result for row convex shape, which is given in [15] in a general setting of polynomial superalgebras. Definition 3.1.…”

“…Then from the definition of straight tableaux, it is straightforward to check that every straight tableau in a skew diagram is a contra-tableau. See also [15,Proposition 4.3]. Lemma 4.3.…”

“…We remark that the multiplicative structure of this ring can be described by the straightening laws, which are in our case essentially Grosshans-Rota-Stein syzygies given in [5]. We refer the readers to [15] for more details.…”

Standard Monomial Theory of RR Varieties

“…From the fact that leading monomials of straight tableaux of a fixed shape are distinct ( [11]), the leading monomial of f should be equal to the leading monomial of T i for some i. Therefore, we have a well defined notion of the leading monomials in(H) for H in R i,m / ker Φ and it is equal to in(T ) for a straight tableau T .…”

Even-primitive vectors in induced supermodules for general linear supergroups and in costandard supermodules for Schur superalgebras

“…According to Theorem 4.4 of [25] (see also Section 5 of [24] or Section 5.7 of [4]), there is a basis for bipermanents of fixed skew shape and content given by bipermanents corresponding to semistandard tableaux. This statement is a generalization of the classical result of [7].…”