A counterexample to a log-concavity conjecture of Brenti
This note records a counterexample to Brenti's 1998 conjecture that the nonzero coefficients of $\widetilde{R}$-polynomials form a log-concave sequence.
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arXiv subjects
Christian Gaetz
Publications and source records attributed to Christian Gaetz.
Bounded Bruhat intervals and affine Coxeter groups
Dyer (1991) proved that, for each fixed length $k$, only finitely many isomorphism types of Bruhat intervals occur in finite Coxeter groups. In this short paper, we prove that this result holds also for affine Coxeter groups, and moreover that this characterizes the affine groups among all infinite Coxeter groups. We also show that the coefficients of $q$ in the Kazhdan-Lusztig polynomials of a Coxeter group are bounded if and only if the group is finite or affine.
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Combinatorial invariance for the coefficient of $q$ in Kazhdan-Lusztig polynomials
We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan--Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture, of Lusztig and of Dyer, also for Bruhat intervals of length at most $6$. We also prove the Gabber--Joseph conjecture for the second-highest $Ext$ group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group, and of the numbers of frozen and of mutable variables in the cluster structure on Richardson varieties.
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Billey-Postnikov posets, rationally smooth Schubert varieties, and Poincar\'e duality
Billey-Postnikov (BP) decompositions govern when Schubert varieties $X(w)$ decompose as bundles of smaller Schubert varieties. We further develop the theory of BP decompositions and show that, in finite type, they can be recognized by pattern conditions and are indexed by the order ideals of a poset $\mathsf{bp}(w)$ that we introduce; we conjecture that this holds in any Coxeter group. We then apply BP decompositions to show that, when $X(w)$ is rationally smooth and $W$ simply laced, the Schubert structure constants $c_{uv}^w$ satisfy a triangularity property, yielding a canonical involution on the Schubert cells of $X(w)$ respecting Poincar\'{e} duality. We also classify the rationally smooth Bruhat intervals in finite type (other than $E$) which admit generalized Lehmer codes, answering questions and conjectures of Billey-Fan-Losonczy, Bolognini-Sentinelli, and Bishop-Mili\'{c}evi\'{c}-Thomas. Finally, we show that rationally smooth Schubert varieties in infinite type need not have Grassmannian BP decompositions, disproving conjectures of Richmond-Slofstra and Oh-Richmond.
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Clasped web bases from hourglass plabic graphs
G.-Pechenik-Pfannerer-Striker-Swanson applied hourglass plabic graphs to construct web bases for spaces of tensor invariants of fundamental representations of $U_q(\mathfrak{sl}_4)$, extending Kuperberg's celebrated basis for $U_q(\mathfrak{sl}_3)$. We give several combinatorial characterizations of basis webs in the kernel of the projection to invariants in a tensor product of arbitrary (type $1$) irreducibles. We apply this to show that the nonzero images of basis webs form a basis (a property shared with Lusztig's dual canonical basis) yielding distinguished clasped web bases for each such tensor product.
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Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras
We study a twisted version of Fraser, Lam, and Le's higher boundary measurement map, using face weights instead of edge weights, thereby providing Laurent polynomial expansions, in Pl\"ucker coordinates, for twisted web immanants for Grassmannians. In some small cases, Fraser, Lam, and Le observe a phenomenon they call "web duality'', where web immanants coincide with web invariants, and they conjecture that this duality corresponds to transposing the standard Young tableaux that index basis webs. We show that this duality continues to hold for a large set of $\text{SL}_3$ and $\text{SL}_4$ webs. Combining this with our twisted higher boundary measurement map, we recover and extend formulas of Elkin-Musiker-Wright for twists of certain cluster variables. We also provide evidence supporting conjectures of Fomin-Pylyavskyy as well as one by Cheung-Dechant-He-Heyes-Hirst-Li concerning classification of cluster variables of low Pl\"ucker degree in $\mathbb{C}[\text{Gr}(3,n)]$.
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Canonical forms of polytopes from adjoints
Projectivizations of pointed polyhedral cones $C$ are positive geometries in the sense of Arkani-Hamed, Bai, and Lam. Their canonical forms look like $$ \Omega_C(x)=\frac{A(x)}{B(x)} dx, $$ with $A,B$ polynomials. The denominator $B(x)$ is just the product of the linear equations defining the facets of $C$. We will see that the numerator $A(x)$ is given by the adjoint polynomial of the dual cone $C^{\vee}$. The adjoint was originally defined by Warren, who used it to construct barycentric coordinates in general polytopes. Confirming the intuition that the job of the numerator is to cancel unwanted poles outside the polytope, we will see that the adjoint is the unique polynomial of minimal degree whose hypersurface contains the residual arrangement of non-face intersections of supporting hyperplanes of $C$.
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The BBDVW Conjecture for Kazhdan-Lusztig polynomials of lower intervals
Blundell, Buesing, Davies, Veli\v{c}kovi\'c, and Williamson (BBDVW) introduced the notion of a hypercube decomposition of an interval in Bruhat order. They conjectured a recursive formula in terms of this structure which, if shown for all intervals, would imply the Combinatorial Invariance Conjecture of Lusztig and Dyer, for Kazhdan-Lusztig polynomials of the symmetric group. In this article, we prove implications between the BBDVW Conjecture and several other recurrences for hypercube decompositions, under varying hypotheses, which have appeared in the recent literature. As an application, we prove the BBDVW Conjecture for lower intervals $[e,v]$, the first non-trivial class of intervals for which it has been established.
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Poset topology, moves, and Bruhat interval polytope lattices
We study the poset topology of lattices arising from orientations of 1-skeleta of directionally simple polytopes, with Bruhat interval polytopes $Q_{e,w}$ as our main example. We show that the order complex $\Delta ((u,v)_w)$ of an interval therein is homotopy equivalent to a sphere if $Q_{u,v}$ is a face of $Q_{e,w}$ and is otherwise contractible. This significantly generalizes the known case of the permutahedron. We also show that saturated chains from $u$ to $v$ in such lattices are connected, and in fact highly connected, under moves corresponding to flipping across a 2-face. When $w$ is a Grassmannian permutation, this implies a strengthening of the restriction of Postnikov's move-equivalence theorem to the class of BCFW bridge decomposable plabic graphs.
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Interlacing triangles, Schubert puzzles, and graph colorings
We show that interlacing triangular arrays, introduced by Aggarwal-Borodin-Wheeler to study certain probability measures, can be used to compute structure constants for multiplying Schubert classes in the $K$-theory of Grassmannians, in the cohomology of their cotangent bundles, and in the cohomology of partial flag varieties. Our results are achieved by establishing a splitting lemma, allowing for interlacing triangular arrays of high rank to be decomposed into arrays of lower rank, and by constructing a bijection between interlacing triangular arrays of rank 3 with certain proper vertex colorings of the triangular grid graph that factors through generalizations of Knutson-Tao puzzles. Along the way, we prove one enumerative conjecture of Aggarwal-Borodin-Wheeler and disprove another.
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The commutant of divided difference operators, Klyachko's genus, and the comaj statistic
In [Hamaker-Pechenik-Speyer-Weigandt, Nenashev, Pechenik-Weigandt] are studied certain operators on polynomials and power series that commute with all divided difference operators $\partial_i$. We introduce a second set of "martial" operators {\martial_i} that generate the full commutant, and show how a Hopf-algebraic approach naturally reproduces the operators $\xi^\nu$ from [Nenashev]. We then pause to study Klyachko's homomorphism $H^*(Fl(n)) \to H^*($the permutahedral toric variety$)$, and extract the part of it relevant to Schubert calculus, the "affine-linear genus''. This genus is then re-obtained using Leibniz combinations of the {\martial_i}. We use Nadeau-Tewari's $q$-analogue of Klyachko's genus to study the equidistribution of $\ell$ and comaj on $[n]\choose k$, generalizing known results on $S_n$.
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A note on Combinatorial Invariance of Kazhdan--Lusztig polynomials
We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the symmetric group. This conjecture has the advantage of being combinatorial in nature. The appendix by Grant T. Barkley and Christian Gaetz discusses the related notion of double hypercubes and proves an analogous conjecture for these in the case of co-elementary intervals.
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On combinatorial invariance of parabolic Kazhdan-Lusztig polynomials
We show that the Combinatorial Invariance Conjecture for Kazhdan-Lusztig polynomials due to Lusztig and to Dyer, its parabolic analog due to Marietti, and a refined parabolic version that we introduce, are equivalent. We use this to give a new proof of Marietti's conjecture in the case of lower Bruhat intervals and to prove several new cases of the parabolic conjectures.
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Web bases in degree two from hourglass plabic graphs
Webs give a diagrammatic calculus for spaces of $U_q(\mathfrak{sl}_r)$-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such $U_q(\mathfrak{sl}_4)$-web bases. Separately, Fraser introduced a web basis for Pl\"{u}cker degree two representations of arbitrary $U_q(\mathfrak{sl}_r)$. Here, we show that Fraser's basis agrees with that predicted by the hourglass plabic graph framework and give an intrinsic characterization of the resulting webs. A further compelling feature with many applications is that our bases exhibit rotation-invariance. Together with the results of our earlier paper, this implies that hourglass plabic graphs give a uniform description of all known rotation-invariant $U_q(\mathfrak{sl}_r)$-web bases. Moreover, this provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions. As a part of our argument, we develop properties of square faces in arbitrary hourglass plabic graphs, a key step in our program towards general $U_q(\mathfrak{sl}_r)$-web bases.
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Rotation-invariant web bases from hourglass plabic graphs
Webs give a diagrammatic calculus for spaces of tensor invariants. We introduce hourglass plabic graphs as a new avatar of webs, and use these to give the first rotation-invariant $U_q(\mathfrak{sl}_4)$-web basis, a long-sought object. The characterization of our basis webs relies on the combinatorics of these new plabic graphs and associated configurations of a symmetrized six-vertex model. We give growth rules, based on a novel crystal-theoretic technique, for generating our basis webs from tableaux and we use skein relations to give an algorithm for expressing arbitrary webs in the basis. We also discuss how previously known rotation-invariant web bases can be unified in our framework of hourglass plabic graphs.
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Promotion permutations for tableaux
In our companion paper, we develop a new $SL_4$-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. A guiding principle behind our construction is that the long cycle $(12\ldots n) \in \mathfrak{S}_n$ should act by rotation of webs. Moreover, the bijection between webs and tableaux should intertwine rotation with the promotion action on tableaux. In this paper, we develop necessary notions of promotion permutations and promotion matrices, which are new even for standard tableaux. To support inductive arguments in the companion paper, we must however work in the more general setting of fluctuating tableaux, which we introduce and which subsumes many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, and (semi)standard tableaux. Therefore, we also give here a full development of the basic combinatorics and representation theory of fluctuating tableaux.
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Combinatorial invariance for Kazhdan-Lusztig $R$-polynomials of elementary intervals
We adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veli\v{c}kovi\'{c}-Williamson to prove the Combinatorial Invariance Conjecture for Kazhdan-Lusztig $R$-polynomials in the case of elementary intervals in $S_n$. This significantly generalizes the main previously-known case of the conjecture, that of lower intervals.
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On the minimal power of $q$ in a Kazhdan-Lusztig polynomial
For $w$ in the symmetric group, we provide an exact formula for the smallest positive power $q^{h(w)}$ appearing in the Kazhdan-Lusztig polynomial $P_{e,w}(q)$. We also provide a tight upper bound on $h(w)$ in simply-laced types, resolving a conjecture of Billey-Postnikov from 2002.
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