Subword complexes in Coxeter groups

Knutson, Allen; Miller, Ezra

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

Cited by 126 publications

(152 citation statements)

References 16 publications

Supporting

Mentioning

149

Contrasting

Unclassified

Order By: Relevance

“…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%

See 3 more Smart Citations

Gröbner geometry of Schubert polynomials

2005

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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%

See 2 more Smart Citations

Gröbner geometry of Schubert polynomials

2005

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.

Self Cite

View full text Add to dashboard Cite

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].

show abstract

Antidiagonal initial ideals

Graduate Texts in Mathematics

View full text Add to dashboard Cite

Subword complexes in Coxeter groups

Cited by 126 publications

References 16 publications

Gröbner geometry of Schubert polynomials

Gröbner geometry of Schubert polynomials

Alternating formulas for K -theoretic quiver polynomials

Antidiagonal initial ideals

Contact Info

Product

Resources

About