Related papers: Exploring a Delta Schur Conjecture

We prove that the symmetric function $\Delta'_{e_{k-1}}e_n$ appearing in the Delta Conjecture can be obtained from the symmetric function in the Rational Shuffle Theorem by applying a Schur skewing operator. This generalizes a formula by…

We conjecture a formula for the symmetric function $\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$ in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr…

For any Schur function $s_{\nu}$, the associated {\em delta operator} $\Delta'_{s_{\nu}}$ is a linear operator on the ring of symmetric functions which has the modified Macdonald polynomials as an eigenbasis. When $\nu = (1^{n-1})$ is a…

We prove the Extended Delta Conjecture of Haglund, Remmel, and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta' _{e_k} e_{n}$, where $\Delta' _{e_k}$ and $\Delta_{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary…

In (Haglund, Remmel, Wilson 2018) Haglund, Remmel and Wilson introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $\Delta'_{e_{n-k-1}} e_n$ in terms of rise-decorated or…

We prove the cases q=0 and t=0 of the generalized Delta conjecture of Haglund, Remmel and Wilson involving the symmetric function $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$. Our theorem generalizes recent results by Garsia, Haglund, Remmel and…

We conjecture two combinatorial interpretations for the symmetric function $\Delta_{e_k} e_n$, where $\Delta_f$ is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations…

In [The Delta Conjecture, Trans. Amer. Math. Soc., to appear] Haglund, Remmel, Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This…

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…

The Shuffle Theorem of Carlsson and Mellit gives a combinatorial expression for the bigraded Frobenius characteristic of the ring of diagonal harmonics, and the Delta Conjecture of Haglund, Remmel and the second author provides two…

We provide a combinatorial interpretation of the symmetric function $\left.\Theta_{e_k}\Theta_{e_l}\nabla e_{n-k-l}\right|_{t=0}$ in terms of segmented Smirnov words. The motivation for this work is the study of a diagonal coinvariant ring…

We prove a compositional refinement of the Delta conjecture (rise version) of Haglund, Remmel and Wilson (2018) for $\Delta_{e_{n-k-1}}'e_n$ which was stated by D'Adderio, Iraci and Vanden Wyngaerd (2020) in terms of Theta operators.

We prove the Schr\"oder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck…

We define a module that is an extension of the diagonal harmonics and whose graded Frobenius characteristic is conjectured to be the symmetric function expression which appears in `the Delta conjecture' of Haglund, Remmel and Wilson…

We prove that $\omega \Delta'_{e_{k}}e_n|_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all…

Gorsky and Negut introduced operators $Q_{m,n}$ on symmetric functions and conjectured that, in the case where $m$ and $n$ are relatively prime, the expression ${Q}_{m,n}(1)$ is given by the Hikita polynomial ${H}_{m,n}[X;q,t]$. Later,…

The shuffle conjecture of Haglund et al. expresses the symmetric function $\nabla e_n$ as a sum over labeled Dyck paths. Here $\nabla$ is an operator on symmetric functions defined in terms of its diagonal action on the basis of modified…

We define a number of related combinatorial objects, each of which possesses a surprising symmetry. We include several applications such as a combinatorial explanation for certain fixed points of the involution $\omega$ on the ring of…

The modified Macdonald polynomials introduced by Garsia and Haiman (1996) have many remarkable combinatorial properties. One such class of properties involves applying the $\nabla$ operator of Bergeron and Garsia (1999) to basic symmetric…

We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both [Haglund 2004] and [Aval et al. 2014]. This settles in particular the cases…