A remarkable formula for counting nonintersecting lattice paths in a ladder with respect to turns

Krattenthaler, Christian; Prohaska, M.

doi:10.1090/s0002-9947-99-01884-x

Cited by 16 publications

(31 citation statements)

References 21 publications

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“…So the formula of 6.9 should be regarded as an "accident" while the combinatorial description of 6.8 holds more generally, for instance, for algebras defined by 1-cogenerated ideals. On the other hand Krattenthaler and Prohaska [54] were able to show that the same "accident" takes place if the paths are restricted to certain subregions called one-sided ladders. This proves a conjecture of Conca and Herzog on the Hilbert series of one-sided ladder determinantal rings; see [24].…”

Section: E(k[x ]/Imentioning

confidence: 96%

Gröbner Bases and Determinantal Ideals

Bruns

Conca

2003

Commutative Algebra, Singularities and Computer Algebra

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Section: E(k[x ]/Imentioning

confidence: 96%

Gröbner Bases and Determinantal Ideals

Bruns

Conca

2003

Commutative Algebra, Singularities and Computer Algebra

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“…, t) in certain vexillary cases; e.g. see [CH94,KP99,Gho02]. Theorem B provides a new proof that Schubert varieties X w ⊆ Fℓ n in the flag manifold are Cohen-Macaulay.…”

Section: Ladder Determinantal Idealsmentioning

confidence: 99%

Gröbner geometry of Schubert polynomials

2005

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Given a permutation w ∈ S n , we consider a determinantal ideal I w whose generators are certain minors in the generic n × n matrix (filled with independent variables). Using 'multidegrees' as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal I w :• variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials;• a Gröbner basis consisting of minors in the generic n × n matrix;• the Stanley-Reisner simplicial complex of the initial ideal in terms of known combinatorial diagrams [FK96], [BB93] associated to permutations in S n ; and• a procedure inductive on weak Bruhat order for listing the facets of this complex.We show that the initial ideal is Cohen-Macaulay, by identifying the StanleyReisner complex as a special kind of "subword complex in S n ", which we define generally for arbitrary Coxeter groups, and prove to be shellable by giving an explicit vertex decomposition. We also prove geometrically a general positivity statement for multidegrees of subschemes. Our main theorems provide a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. More precisely, we apply these theorems to:• define a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric reasons;*AK was partly supported by the Clay Mathematics Institute, Sloan Foundation, and NSF. EM was supported by the Sloan Foundation and NSF. ALLEN KNUTSON AND EZRA MILLER• rederive from a topological perspective Fulton's Schubert polynomial formula for universal cohomology classes of degeneracy loci of maps between flagged vector bundles;• supply new proofs that Schubert and Grothendieck polynomials represent cohomology and K-theory classes on the flag manifold; and• provide determinantal formulae for the multidegrees of ladder determinantal rings.The proofs of the main theorems introduce the technique of "Bruhat induction", consisting of a collection of geometric, algebraic, and combinatorial tools, based on divided and isobaric divided differences, that allow one to prove statements about determinantal ideals by induction on weak Bruhat order.

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“…While explaining what is behind the individual steps of the above algorithm, we illustrate each of them by the running example in which A = (0, 1), S 1 = (2, 2), S 2 = (4, 3), S 3 = (2, 5), S 4 = (8, 9), S 5 = (10, 10), S 6 = (11, 11), T 1 = (4, 1), T 2 = (5, 1), T 3 = (6, 1), T 4 = (5, 2), T 5 = (5, 5), T 6 = (5, 6), T 7 = (8, 7), T 8 = (11, 9), T 9 = (13, 10), B = (12,14); see Figure 8. Clearly, there is nothing to be said about…”

Section: Sketch Of Proofmentioning

confidence: 99%

“…Abhyankar and Kulkarni [2] have shown that the Hilbert function of ladder determinantal rings coincides with the Hilbert polynomial at all nonnegative integers. For more work on ladder determinantal rings, see [4,6,7,9,13,14,15,16,20].…”

Section: Introductionmentioning

confidence: 99%