Vic Reiner- open conjectures, questions, problems
I've accumulated a few conjectures, questions and problems over the years.
Here are some of my favorites that are unresolved,
as far as I am aware, listed roughly in reverse chronological order.
- (with B. Rhoades)
Rationality of the q-Ehrhart series and finite generation of the harmonic algebra.
See Conjectures 1.1 and 5.5 in here.
- (with A. Adams)
Resolutions of simplicial poset face rings over their canonical parameters.
See Conjecture 6.1 in here.
- (with G. Tudose and the 2020 Polymath Jr. group)
Hilbert series for subalgebra filtrations of cohomology of ordinary and Lagrangian Grassmannians.
See Conjecture 1 in here,
and Conjecture 1.4 in here.
- (with J. Lewis)
"Root system circuits are acutely disconnected".
Is there a non case-by-case proof of Lemma 1.2
in here,
which asserts this fact about finite real reflection groups:
any set of roots forming a minimal linear dependence with positive coefficients has a disconnected graph of pairwise acuteness.
- (with J. Lewis and D. Stanton)
The invariants of GL_{n}(F_{q}) mod Frobenius powers conjecture.
See Conjecture 1.1 in here.
- (with D. Stanton and D. White)
Are there insightful, non-brute force proofs of the cyclic sieving phenomena on ...
- triangulations of a convex n-gon under n-fold rotation?
- alternating sign matrices under 4-fold rotation?
See the discussion at the end here.
- (with J. Lewis and D. Stanton)
A q-analogue of Hurwitz's formula.
Is there a short proof of Theorem 1.1 in here,
showing there are (q^{n}-1)^{n-1} factorizations of a Singer cycle in
GL_{n}(F_{q}) into n reflections,
similar to proofs that n-cycles have n^{n-2} factorizations into n-1 transpositions?
- (with D. Armstrong and B. Rhoades)
The parking space conjecture.
See the Main Conjecture from Section 2.6 in here.
- (with F. Saliola and V. Welker)
The symmetrized k-subset-to-top shuffling operators commute.
Is there a conceptual, non-inductive proof of Theorem 1.1 in
here?
- (with Y. Roichman)
Diameter of the graph of reduced words for the longest element in type D_{n}.
Is the lower bound given by the number of codimension two intersection subspaces (see Table 1.1 here) tight or not?
- (with A. Miller)
Characteristic polynomials det(UD-tI) of up-down maps in differential posets have a Smith form over Z[t].
See Conjecture 1.1 in here.
- The number of braid relations in reduced words for the longest element of S_{n} approaches Poisson with mean 1.
See Conjecture 2 in here
- (with A. Duval)
The generalized Grone-Merris conjecture.
See Conjecture 1.2 in here, which says that
in a d-dimensional simplicial complex,
the top dimensional Laplacian spectrum is majorized by the conjugate partition to the sequence of vertex degrees with respect to the d-simplices.
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