Subword complexes in Coxeter groups

Knutson, Allen; Miller, Ezra

doi:10.1016/s0001-8708(03)00142-7

Cited by 124 publications

(152 citation statements)

References 16 publications

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“…However, a more detailed analysis would take us too far afield, so that we have chosen to develop the theory of subword complexes in Coxeter groups more fully elsewhere [KnM04]. There, we show that subword complexes are balls or spheres, and calculate their Hilbert series for applications to Grothendieck polynomials.…”

Section: Subword Complexes In Coxeter Groupsmentioning

confidence: 96%

“…But the Alexander inversion formula [KnM04] implies that G w (1 − x) is the K-polynomial of J w , given that G w (x) is the K-polynomial of k[z]/J w as in Theorem A. Therefore, G w (1 − x) must alternate as in (3), if the CohenMacaulayness in Theorem B holds.…”

Section: Positive Formulae For Schubert Polynomialsmentioning

confidence: 99%

“…We carry out this procedure in Section 2.1 using multidegrees, for which the required technology is developed in Section 1.7. The analogous K-theoretic formula, which additionally involves nonreduced pipe dreams, requires more detailed analysis of subword complexes (Definition 1.8.1), and therefore appears in [KnM04]. Example 1.5.2.…”

Section: Gröbner Geometrymentioning

confidence: 99%

“…Readers wishing to see the determinantal expression in its full glory can check [Ful92] for a brief introduction to multi-Schur polynomials, or [Mac91] for much more. It would be desirable to make the Hilbert series in Theorem B just as explicit in closed form, given that combinatorial formulae are known (see [KnM04] for details and references): …”

Section: Corollary 241 the Z 2n -Graded Multidegree Of A Ladder Dementioning

confidence: 99%

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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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Section: Subword Complexes In Coxeter Groupsmentioning

confidence: 96%

Section: Positive Formulae For Schubert Polynomialsmentioning

confidence: 99%

Section: Gröbner Geometrymentioning

confidence: 99%

Section: Corollary 241 the Z 2n -Graded Multidegree Of A Ladder Dementioning

confidence: 99%

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Gröbner geometry of Schubert polynomials

2005

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“…This technique is combined with those developed in [KM03b] for dealing with nonreduced subwords of reduced expressions for permutations.…”

Section: Introductionmentioning

confidence: 99%

Alternating formulas for K -theoretic quiver polynomials

Miller¹

2005

Duke Math. J.

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Abstract. The main theorem here is the K-theoretic analogue of the cohomological 'stable double component formula' for quiver functions in [KMS03]. This K-theoretic version is still in terms of lacing diagrams, but nonminimal diagrams contribute terms of higher degree. The motivating consequence is a conjecture of Buch on the sign-alternation of the coefficients appearing in his expansion of quiver K-polynomials in terms of stable Grothendieck polynomials for partitions [Buc02a].

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Antidiagonal initial ideals

Graduate Texts in Mathematics

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Subword complexes in Coxeter groups

Cited by 124 publications

References 16 publications

Gröbner geometry of Schubert polynomials

Gröbner geometry of Schubert polynomials

Alternating formulas for K -theoretic quiver polynomials

Antidiagonal initial ideals

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