467 research outputs found
Lagrangian Geometry Towards Type AxC Schubert Calculus
No full text59 pagesKnutson and Zinn-Justin gave a geometric explanation of the Knutson-Tao puzzle rule for computing Schubert calculus of certain partial flag varieties in type A . The geometry is based on Lagrangian correspondences between Nakajima quiver varieties. We extend Knutson and Zinn-Justin's construction to triple products of Nakajima quiver varieties. By defining a compatible Z/2Z action on our quiver varieties, we furthermore construct Lagrangian correspondences between cotangent bundles of Grassmannians and symplectic Grassmannians. This is the first, and hardest, step towards a puzzle rule for computing the product of symplectic Grassmannian Schubert classes by Grassmannian Schubert classes
Global representation ring and Knutson Index
No full textGlobal representation rings were discovered by Sarah Witherspoon in 1995 and the Knutson Index was introduced by the second author in 2022. In the present paper we introduce the Knutson Index for general commutative rings and study it for Burnside rings and global representation rings. We also introduce the global table of a finite group, that encompasses both the character table and the Burnside table of marks. We discuss what properties of a group can be recovered from its global table
Milo Knutson: The Man and the Myth
No full textMilo Knutson came to LaCrosse in 1917 and established
himself as a news broadcaster. Prior to his arrival he
had never held either an elective or appointed public
position. Yet, he eventually sought and won the mayoralty
of LaCrosse, an office he maintained from 1955 to 1965.
His rise in local popularity can be attributed
to several personal and community factors. Knutson's
dynamic personality, oratorical skills, and political
opportunism were his most important public assets. The James
Christie Case, the Evelyn Hartley Case, conflict among
some community leaders, and the conservative temper of
LaCrosse Citizens provided a receptive social and political
climate for an ambitious person, such as Milo Knutson, who
held conservative views.
As a candidate in the 1955 election, Milo Knutson accused
community leaders of incompetence. Subsequently, he made
political pledges to correct what he considered to be
existing abuses within the city. Apparently anticipating
reforms, the majority of the community supported Knutson.
But the new mayor found himself in the same political situation,
that, as a candidate, he had accused his predecesors of occupying. Knutson failed to fulfill his political pledges
to reform the police department and to reconstruct city
government. Once in office, he exhibited the same weaknesses
he had denounced in his predecessors.
The issue of public education became an effective
"tool" which Knutson manipulated for his popular and political
gain. Throughout his administration, his critics charged
that public education regressed because of insufficient money,
too few teachers, and inadequate school facilities.
The author concludes that Milo Knutson, seeking a
public career, took advantage of a timely series of events
to win election as mayor of LaCrosse. His administration
was, however, a general failure in terms of his ability to
fulfill important campaign promises and in terms of his
support for important institutions within the community
Toric surfaces with Kazhdan-Lusztig atlases
No full textA Kazhdan-Lusztig atlas, introduced by He, Knutson and Lu, on a stratified variety (V,Y) is a way of modeling the stratification Y of V locally using the stratification of Kazhdan-Lusztig varieties. We are interested in classifying smooth toric surfaces with Kazhdan-Lusztig atlases. This involves finding a degeneration of V to a union of Richardson varieties in the flag variety H/B_H of some Kac-Moody group H. We determine which toric surfaces have a chance at having a Kazhdan-Lusztig atlas by looking at their moment polytopes, then describe a way to find a suitable group H. More precisely, we find that (up to equivalence) there are 19 or 20 broken toric surfaces admitting simply-laced atlases, and that there are at most 7543 broken toric surfaces where H is any Kac-Moody group
Growth Diagrams from Polygons in the Affine Grassmannian
No full textWe introduce growth diagrams arising from the geometry of the affine Grassmannian for . These affine growth diagrams are in bijection with the many components of the polygon space Poly() for a sequence of minuscule weights and the Littlewood--Richardson coefficient. Unlike Fomin growth diagrams, they are infinite periodic on a staircase shape, and each vertex is labeled by a dominant weight of . Letting go to infinity, a dominant weight can be viewed as a pair of partitions, and we recover the RSK correspondence and Fomin growth diagrams within affine growth diagrams. The main combinatorial tool used in the proofs is the -hive of Knutson--Tao--Woodward. The local growth rule satisfied by the diagrams previously appeared in van Leeuwen's work on Littelmann paths, so our results can be viewed as a geometric interpretation of this combinatorial rule. Similar diagrams appeared in the work of Speyer on osculating flags
Subword complexes in Coxeter groups
No full textAbstractLet (Π,Σ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex, as introduced in Knutson and Miller (Ann. of Math. (2) (2003), to appear). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which appeared in Fomin and Kirillov (Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183–190), are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented
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Where Upside Down is Right Side Up: A Study of Ksemendra's Narmamala and His Theory of Aucitya
No full textThis paper examines the relationship between 11th century Sanskrit author Ksemendra’s theory of literary propriety and satire in his work, the Narmamala. In what her professor describes as “an original work of scholarship in the field,” Gomez’s close examination of primary texts and exploration into Sanskrit literary theory led her to an appreciation of Ksemendra’s work not commonly shared by others in the field. Gomez used a range of Library digital resources, including JSTOR, Oskicat, and Melvyl, as well as Berkeley’s expansive collections in the South and Southeast Asian Library. Professor Knutson believes that she “has accomplished something exceedingly rare for an undergraduate, [which] would have been impossible to do this [with] a lesser institution’s collection.
The Artistic Legacy of Georgia O’Keeffe
No full textThe work of American artist, Georgia O’Keeffe (1887-1986), was considered to be unique and revolutionary in her own time, but she ultimately achieved a prominent position in the history of art. The enduring inspiration of this important artist is demonstrated by an investigation of key artistic motifs as well as her legacy via the discussion of artists who have been influenced by her: Alfred Stieglitz, Arthur Dove, Marsden Hartley, and the author, Michelle L. Knutson
Schubert puzzles and integrability I: invariant trilinear forms
No full textThe puzzle rules for computing Schubert calculus on -step flag manifolds,
proven in [Knutson Tao 2003] for -step, in [Buch Kresch Purbhoo Tamvakis
2016] for -step, and conjectured in [Coskun Vakil 2009] for -step, lead
to vector configurations (one vector for each puzzle edge label) that we
recognize as the weights of some minuscule representations. The -matrices of
those representations (which, for -step flag manifolds, involve triality of
) degenerate to give us puzzle formulae for two previously unsolved
Schubert calculus problems: -step flag manifolds and -step flag
manifolds. The -step flag manifolds formula, which involves 151 new
puzzle pieces, implies Buch's correction to the first author's 1999 conjecture
for -step flag manifolds.Comment: v5: misleading sentence in the statement of theorem 2 and missing
pictures in the statement of theorem 3 fixed. no results or proofs changed.
v6: left vs right coset issues fixe
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