On the symmetric group action on rigid disks in a strip

Wawrykow, Nicholas

doi:10.1007/s41468-022-00111-8

Cited by 2 publications

(1 citation statement)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, in the case w = 2 Alpert proved that the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 exhibits a reasonable notion of first-order representation stability, as, for all k, the homology groups H k conf(•, 2) have the structure of a finitely generated FI k+1 -module, generated in degree at most 3k, where FI k+1 is a generalization of FI# [Alp20]. One can use Alpert's results to decompose H k conf(n, 2); Q into a direct sum of irreducible S n -representations as was done in [Waw22a]. Alpert's proof relied on determining a basis for H k conf(n, 2) , and using this basis to show that there are well-defined ways to insert the fundamental class of H 0 conf(1, w) into a class in H k conf(n, 2) to get a class in H k conf(n + 1, 2) .…”

Section: Introductionmentioning

confidence: 99%

Representation Stability for Disks in a Strip

Wawrykow¹

2023

Preprint

View full text Add to dashboard Cite

We consider the ordered configuration space of n open unit-diameter disks in the infinite strip of width w. In the spirit of Arnol'd and Cohen, we provide a finite presentation for the rational homology groups of this ordered configuration space as a twisted algebra. We use this presentation to prove that the ordered configuration space of open unit-diameter disks in the infinite strip of width w exhibits a notion of first-order representation stability similar to Church-Ellenberg-Farb and Miller-Wilson's firstorder representation stability for the ordered configuration space of points in a manifold. In addition, we prove that for large w this disk configuration space exhibits notions of second-(and higher) order representation stability.

show abstract

Section: Introductionmentioning

confidence: 99%

Representation Stability for Disks in a Strip

Wawrykow¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

The topological complexity of the ordered configuration space of disks in a strip

Wawrykow

2024

Proc. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

How hard is it to program n n robots to move about a long narrow aisle such that only w w of them can fit across the width of the aisle? In this paper, we answer that question by calculating the topological complexity of conf ⁡ ( n , w ) \operatorname {conf}(n,w) , the ordered configuration space of n n open unit-diameter disks in the infinite strip of width w w . By studying its cohomology ring, we prove that, as long as n n is greater than w w , the topological complexity of conf ⁡ ( n , w ) \operatorname {conf}(n,w) is 2 n − 2 ⌈ n w ⌉ + 1 2n-2\big \lceil \frac {n}{w}\big \rceil +1 , providing a lower bound for the minimum number of cases such a program must consider.

show abstract

On the symmetric group action on rigid disks in a strip

Cited by 2 publications

References 15 publications

Representation Stability for Disks in a Strip

Representation Stability for Disks in a Strip

The topological complexity of the ordered configuration space of disks in a strip

Contact Info

Product

Resources

About