in Math. Post comments on them. But upon retirement (2018) I am adding some of the latter. Sage Mathematics Software (Version 6.10), The Sage Development Team, http://www.sagemath.org , 2016. An open problem about understanding the coinvariants of the symmetric group action on polynomials in multiple sets of (commuting and anti-commuting) variables. Imp.¹: Importance (Low , Medium , High , Outstanding ) Rec.²: Recommended for undergraduates. To supplement the list you found, DIMACS Open problems for undergraduates, there is The Open Problems Project, the latter not specifically oriented to undergraduates. $\endgroup$ – Joseph O'Rourke Dec 11 '15 at 17:41 Hindman, Neil Topological properties of a random partition of the plane, The online MST of the randomly-weighted complete graph. Itai Benjamini This collection was started as this 2003 PDF document and became a web site Swapping out our Syntax Highlighter, Responding to the Lavender Letter and commitments moving forward. The coefficient of the Schur function in (where is a partition of ) is equal to the coefficient of in the Schur function evaluation . for this article. : /Spencer : Consider p ( v , t ), the probability that a walk beginning from the origin ends at the point v on the d -dimensional integer lattice in time t . A discrete Hammersley process as an extreme case of oriented percolation flows, Local weak convergence of random graphs and networks. 2012. ). Since the permutation matrices are a natural subgroup of , when an irreducible representation is restricted from to it decomposes as a direct sum of irreducible representations. March 19, 2020 by moderator. s[2, 1, 1] + s[2, 2] + 4*s[3, 1] + 3*s[4] An open problem about understanding combinatorially the characters of the symmetric group acting on tensor powers of representations of a reductive group. Jockusch, Carl G. A community blog for the global algebraic combinatorics community to share open problems with one another. If we evaluate the symmetric function at the eigenvalues , this is the value of the character at a permutation of cycle structure . arXiv:1605.06672. Create and edit open problems pages (please contact us and we will set you up an account. and Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular. Usage data cannot currently be displayed. A conjectured compactification Edit: I'm pretty sure this problem is mentioned in Stanley, but it is worth emphasizing that it is still open and that there is a lot of beautiful combinatorics related to it. A representation of is a homomorphism from to . Hindman, Neil but it is quite old and many of the problems are solved. Check if you have access via personal or institutional login, COPYRIGHT: © Cambridge University Press 2003. It only takes a minute to sign up. [closed], http://math.mit.edu/~rstan/pubs/pubfiles/116.pdf, A Littlewood-Richardson rule for two-step flag varieties, Goodbye, Prettify. 2, Cambridge University Press, 1999. One of the oldest standing open problems in algebraic combinatorics is Foulkes' conjecture; for some history and nice reformulations of the problem, see On Foulkes' conjecture by William F. Doran IV in Journal of Pure and Applied Algebra (August 1998), 130 (1), pg. Welcome to the Open Problem Garden, a collection of unsolved problems in mathematics. Abstract views reflect the number of visits to the article landing page. Change ), You are commenting using your Google account. Leader, Imre and What are the most important open problems in algebraic combinatorics? Шкредов, Илья Дмитриевич Open problems concerning all the finite groups. 2010. COMBINATORICS SOME OPEN PROBLEMS Richard P. Stanley 1. Even if the öweightö of an edge (i.e., one plus the number of steps taken between two points) is never incremented beyond 2, this problem is open. Edinburgh Math. Hilbert series of graded Cohen-Macaulay domains, 28 years later? Shortest routes in random proximity networks. Many thanks to our contributors! In the case when , we have the following, which should be a special case of what the answer might look like in general: Proposition. Coding invariant processes on infinite trees. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. 85-98. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Our aim in this paper is to present some of the natural and appealing open problems in the area. The value of is the dimension of the representation. Open Problems in Algebraic Combinatorics May 18-22, 2020 May 17-21, 2021 (rescheduled date tentative) University of Minnesota Organizers: Christine Berkesch, Ben Brubaker, Gregg Musiker, Pavlo Pylyavskyy, Vic Reiner * Views captured on Cambridge Core between
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