The Hilbert Series of Pfaffian Rings

Ghorpade, Sudhir R.; Krattenthaler, Christian

doi:10.1007/978-3-642-18487-1_22

Cited by 12 publications

(22 citation statements)

References 29 publications

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“…It follows that the multiplicity equals the number of these paths. This interpretation generalizes those in [3,5]. We first illustrate by means of examples and then provide a brief justification.…”

Section: Multiplicity Counts Certain Non-intersecting Lattice Pathsmentioning

confidence: 82%

“…This proves (a). Assertion (b) now follows from Proposition 5.1.1 (3). For (c), write p h (α) = (a, a * ), λ = (R, C), and γ = (r, c).…”

Section: 1mentioning

confidence: 90%

“…(2) follows immediately from the definition of σ j . We now prove (3). First suppose β > p h (σ j ) for some β in OE pr j +1 .…”

Section: Proposition 631mentioning

confidence: 96%

“…The points of OR(v) are those that are (strictly) above the diagonal, and the points of N(v) are those that are to the South-West of the poly-line captioned 'boundary of N(v)': we draw the boundary so that points on the boundary belong to N(v). The reader can readily verify that d = 13 and v = (1,2,3,4,6,7,10,11,13,15,18,19,22) We will be considering monomials, also called multisets, in some of these sets. A monomial, as usual, is a subset with each member being allowed a multiplicity (taking values in the non-negative integers).…”

Section: Basic Notationmentioning

confidence: 99%

“…The papers quoted in the previous paragraph and [3] give some answers in the special cases they consider. More interestingly, the recent article [6] gives not only an interpretation of the multiplicity similar to ours but also a closed form formula.…”

mentioning

confidence: 97%

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Hilbert functions of points on Schubert varieties in orthogonal Grassmannians

Raghavan

Upadhyay

2009

J Algebr Comb

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Given a point on a Schubert variety in an orthogonal Grassmannian, we compute the multiplicity, more generally the Hilbert function. We first translate the problem from geometry to combinatorics by applying standard monomial theory. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for Grassmannians and symplectic Grassmannians.As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. A more important application, although it does not appear here but elsewhere, is to the computation of the initial ideal, with respect to certain convenient monomial orders, of the ideal of the tangent cone to the Schubert variety.Taking the Schubert variety to be of a special kind and the point to be the 'identity coset,' our problem specializes to one about Pfaffian ideals, treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a 'generic singularity.'

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