scholarly journals An equivariant basis for the cohomology of Springer fibers

2021 ◽  
Vol 8 (17) ◽  
pp. 481-509
Author(s):  
Martha Precup ◽  
Edward Richmond
Keyword(s):  
Springer Fibers

scholarly journals Combinatorics of Tableau Inversions

10.37236/4932 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Jonathan E. Beagley ◽  
Paul Drube
Keyword(s):  
Betti Number ◽  
Young Tableau ◽  
Betti Numbers ◽  
Young Tableaux ◽  
Standard Tableau ◽  
Specific Standard ◽  
Different Shapes ◽  
Springer Fibers ◽  
Standard Young Tableau ◽  
Fixed Standard

A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard.  An $i$-inverted Young tableau is a row-standard tableau along with precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Finally, we share some the results of a computer program developed to calculate tableaux inversions.

scholarly journals Regular points in affine Springer fibers

2005 ◽  
Vol 53 (1) ◽  
pp. 97-107 ◽  
Author(s):  
Mark Goresky ◽  
Robert Kottwitz ◽  
Robert MacPherson
Keyword(s):  
Springer Fibers

scholarly journals Study of some orthosymplectic Springer fibers

2011 ◽  
Vol 335 (1) ◽  
pp. 83-95
Author(s):  
Séverine Leidwanger ◽  
Nicolas Perrin
Keyword(s):  
Springer Fibers

scholarly journals Inversions of Semistandard Young Tableaux

10.37236/5469 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Paul Drube
Keyword(s):  
Young Tableau ◽  
Betti Numbers ◽  
Young Tableaux ◽  
Closed Formula ◽  
General Shape ◽  
Combinatorial Objects ◽  
Standard Tableau ◽  
Specific Shape ◽  
Springer Fibers ◽  
Standard Young Tableaux

A tableau inversion is a pair of entries from the same column of a row-standard tableau that lack the relative ordering necessary to make the tableau column-standard. An $i$-inverted Young tableau is a row-standard tableau with precisely $i$ inversion pairs, and may be interpreted as a generalization of (column-standard) Young tableaux. Inverted Young tableaux that lack repeated entries were introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, and were later developed as combinatorial objects in their own right by Beagley and Drube. This paper generalizes earlier notions of tableau inversions to row-standard tableaux with repeated entries, yielding an interesting new generalization of semistandard (as opposed to merely standard) Young tableaux. We develop a closed formula for the maximum numbers of inversion pairs for a row-standard tableau with a specific shape and content, and show that the number of $i$-inverted tableaux of a given shape is invariant under permutation of content. We then enumerate $i$-inverted Young tableaux for a variety of shapes and contents, and generalize an earlier result that places $1$-inverted Young tableaux of a general shape in bijection with $0$-inverted Young tableaux of a variety of related shapes.

scholarly journals Real Springer fibers and odd arc algebras

2020 ◽  
Author(s):  
Jens Niklas Eberhardt ◽  
Grégoire Naisse ◽  
Arik Wilbert
Keyword(s):  
Springer Fibers

scholarly journals TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS

2021 ◽  
pp. 1-62
Author(s):  
Zongbin Chen
Keyword(s):  
Fundamental Domain ◽  
Recurrence Relations ◽  
Rational Points ◽  
Orbital Integrals ◽  
Orbital Integral ◽  
Comparison Of Results ◽  
Fundamental Domains ◽  
Springer Fiber ◽  
Springer Fibers

Abstract We explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

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