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Register a new userThe Delta Conjecture at $q=1$
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Marino Romero
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We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of $Delta_{e_k} e_n$ at $q=1$ in terms of the elementary symmetric function basis. We then use a weight-preserving bijection to prove the Delta Conjecture at $q=1$. The method of proof provides a variety of structures which can compute the inner product of $Delta_{e_k} e_n|_{q=1}$ with any symmetric function.
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The distribution of certain Mahonian statistic (called $mathrm{BAST}$) introduced by Babson and Steingr{i}msson over the set of permutations that avoid vincular pattern $1underline{32}$, is shown bijectively to match the distribution of major index over the same set. This new layer of equidistribution is then applied to give alternative interpretations of two related $q$-Stirling numbers of the second kind, studied by Carlitz and Gould. Moreover, extensions to an Euler-Mahonian statistic over ordered set partitions, and to statistics over ordered multiset partitions present themselves naturally. The latter of which is shown to be related to the recently proven Delta Conjecture. During the course, a refined relation between $mathrm{BAST}$ and its reverse complement $mathrm{STAT}$ is derived as well.
The symmetric group $mathfrak{S}_n$ acts on the polynomial ring $mathbb{Q}[mathbf{x}_n] = mathbb{Q}[x_1, dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $mathfrak{S}_n$-invariant polynomials with vanishing constant term. The quotient $R_n = frac{mathbb{Q}[mathbf{x}_n]}{I_n}$ is called the coinvariant algebra. The coinvariant algebra $R_n$ has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization $I_{n,k} subseteq mathbb{Q}[mathbf{x}_n]$ of the ideal $I_n$ indexed by two positive integers $k leq n$. The corresponding quotient $R_{n,k} := frac{mathbb{Q}[mathbf{x}_n]}{I_{n,k}}$ carries a graded action of $mathfrak{S}_n$ and specializes to $R_n$ when $k = n$. We generalize many of the nice properties of $R_n$ to $R_{n,k}$. In particular, we describe the Hilbert series of $R_{n,k}$, give extensions of the Artin and Garsia-Stanton monomial bases of $R_n$ to $R_{n,k}$, determine the reduced Grobner basis for $I_{n,k}$ with respect to the lexicographic monomial order, and describe the graded Frobenius series of $R_{n,k}$. Just as the combinatorics of $R_n$ are controlled by permutations in $mathfrak{S}_n$, we will show that the combinatorics of $R_{n,k}$ are controlled by ordered set partitions of ${1, 2, dots, n}$ with $k$ blocks. The {em Delta Conjecture} of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of $R_{n,k}$ is (up to a minor twist) the $t = 0$ specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded $mathfrak{S}_n$-module $V_{n,k}$ whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module $R_{n,k}$ solves this problem in the specialization $t = 0$.
A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. v{C}erny conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$. We introduce the notion of aperiodically $1-$contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which the v{C}erny conjecture holds for aperiodically $1-$contracting automata. As a special case, we prove some results for circular automata.
We introduce a family of varieties $Y_{n,lambda,s}$, which we call the $Delta$-Springer varieties, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring $H^*(Y_{n,lambda,s})$ and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induced Specht modules. The $lambda=(1^k)$ case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at $t=0$. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety $Y_{n,lambda}$ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud-Saltman rank variety and diagonal matrices.
The Markoff injectivity conjecture states that $wmapstomu(w)_{12}$ is injective on the set of Christoffel words where $mu:{mathtt{0},mathtt{1}}^*tomathrm{SL}_2(mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $mu_q$ of $mu$ such that $mu_{qto1}(w)=mu(w)$ is the Markoff number associated to the Christoffel word $w$. We show that for every $q>0$, the map ${mathtt{0},mathtt{1}}^*tomathbb{Z}[q]$ defined by $wmapstomu_q(w)_{12}$ is injective over the language $mathcal{L}(s)$ of a balanced sequence $sin{mathtt{0},mathtt{1}}^mathbb{Z}$. The proof is based on new equivalent definitions of balanced sequences.
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