arXiv 1508.062392015-08math.RTmath.CO

A proof of the shuffle conjecture

Erik Carlsson, Anton Mellit

备注：: some proofs are expanded. Accepted in JAMS

英文摘要：

We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space $V_*$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $\mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $\tildeÅ$ acting on this space, and interpret the right generalization of the $\nabla$ using an involution of the algebra which is antilinear with respect to the conjugation $(q,t)\mapsto (q^{-1},t^{-1})$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A Proof the Functional Equation Conjecture

In the early 2000's the first and second named authors worked for a period of six years in an attempt of proving the Compositional Shuffle Conjecture [1]. Their approach was based on the discovery that all the Combinatorial properties predicted by the Compositional Shuffle Conjecture remain valid for each family of Parking Functions with prescribed diagonal cars. The validity of this property was reduced to the proof of a functional equation satisfied by a Catalan family of univariate polynomials. The main result in this paper is a proof of this functional equation. The Compositional Shuffle C…

2018-06-08math.COmath.RT

A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn's formula

Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's formula when interpreted in the framework of signatures of paths.

2021-03-15math.COmath.PRmath.RA

Schedules and the Delta Conjecture

In a recent preprint, Carlsson and Oblomkov (2018) obtain a long sought after monomial basis for the ring $\operatorname{DR}_n$ of diagonal coinvariants. Their basis is closely related to the "schedules" formula for the Hilbert series of $\operatorname{DR}_n$ which was conjectured by the first author and Loehr (2005) and first proved by Carlsson and Mellit (2018), as a consequence of their proof of the famous Shuffle Conjecture. In this article we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by the first author, Remmel and Wilson (2018)…

2020-03-31math.COmath.RT

Shuffle algebras for quivers and R-matrices

We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.

2022-03-14math.QAmath.RT

扩展Delta猜想的谷型版本

The valley version of the Extended Delta Conjecture

本文提出扩展Delta猜想的谷型版本，并通过研究扩展有序多重集划分的组合学，证明当t或q等于0时该猜想与上升版本等价。

2019-07-09math.COmath.RT

费金四面体

The Feigin Tetrahedron

本文的第一个目标是将著名的带量子群的Feigin同态推广到量子多项式代数。更确切地说，我们定义了广义Feigin同态从一个量子shuffle代数到量子多项式代数，它扩展了经典Feigin同态沿嵌入的量子群到所述量子shuffle代数。在Berenstein和作者最近的一项工作中，从一个有价值的代数的对偶Hall-Ringel代数到量子多项式代数的Feigin同态的类似扩展被定义。为了将这些构造联系起来，我们建立了一个同态，称为量子洗牌字符，从对偶Hall-Ringel代数到量子洗牌代数，其中涉及广义Feigin同态。这些构造可以用一个从量子群开始到量子多项式代数结束的映射的交换四面体来紧描述。这个项目的第二个目标是更好地理解可斜对称化量子团簇代数的对偶正则基猜想。在可对称化的类型中，已知对偶正则基元不需要具有正的乘法结构常数，而这仍然被怀疑适用于斜对称化的量子簇代数。我们对可对称化类型提出了另一个猜想：簇单项式应该对应于KLR代数的不可约特征标。事实上，本说明的主要猜想将建立这种“KLR猜想”的非循环斜对称量子簇代数：也就是说，我们猜想，刚性表示的图像下的量子洗牌字符给KLR代数的不可约字符。

2015-03-19math.QAmath.RT

更多相关论文收起更多论文

折叠洗牌代数与扭曲 $q$-特征

Folding shuffle algebras and twisted $q$-characters

École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Simion Stoilow Institute of Mathematics (IMAR)（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；Faculty of Mathematics, University of Vienna（维也纳大学数学系）

通过引入折叠洗牌代数，证明了量子非扭曲仿射代数模的某些 $q$-特征与其扭曲对应物相等的Hernandez猜想，并将结果推广到具有自同构的任意箭图，特别地定义并描述了扭曲量子环面代数。

2026-06-16math.QAmath.RT

BPS李代数、反常滤过与洗牌代数

BPS Lie algebras, perverse filtrations and shuffle algebras

École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Simion Stoilow Institute of Mathematics (IMAR)（西蒙·斯托伊洛瓦数学研究所）

通过将上同调Hall代数上的反常滤过与多项式的极限条件关联，显式描述了零势能箭图的BPS李代数，并部分推广到任意势能情形。

2026-06-19math.AGmath.QAmath.RT

PBWD bases and shuffle algebra realizations for $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ and their integral forms

We construct a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the quantum loop algebras $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ in the new Drinfeld realizations. In the 2-parameter case, this proves Theorem 3.11 of [Hu-Rosso-Zhang] (stated in loc. cit. without a proof), while in the super case it proves a conjecture of [Zhang]. The main ingredient in our proofs is the interplay between those quantum loop algebras and the corresponding shuffle algebras, which are trigonometric counterparts of the elliptic shuffle algebras of Feigin-Odesskii. …

2021-04-09math.RT

Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems

We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras, thus generalizing its rational version of arXiv:2104.14518 and the type A construction of arXiv:1811.12137. As an application, this allows us to construct large families of commuting and q-commuting difference operators, in particular, providing a convenient approach to the Q-systems where it proves a conjecture of arXiv:1704.00154.

2023-01-04math-phmath.MPmath.QA

torus link homology 的 shuffle 定理

Shuffle theorem for torus link homology

Seoul National University（首尔大学）；Sungkyunkwan University（成均馆大学）

本文证明了椭圆Hall代数与torus链接的Khovanov-Rozansky同调之间的关系，并解决了关于停车函数生成函数的猜想。

2026-02-06math.COmath.GTmath.QA

The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called "shuffles", and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps: first we reinterpret the shuffles equations in terms of Frobenius twists of symmetr…

2018-06-14math.AGmath.RT

关于Iwahori--Hecke代数的shuffle元素的注记

Remarks on the shuffle elements of Iwahori--Hecke algebras

Beijing Normal University at Zhuhai（北京师范大学珠海校区）

本文证明当n≥4时Iwahori--Hecke代数中长度为1和2的shuffle元素非交换，否证了Doikou猜想，并证明所有左乘算子矩阵具有q-对称性。

2026-09-22math.RT

Khovanov-Rozansky同调与高阶Catalan序列

Khovanov-Rozansky homology and higher Catalan sequences

我们给出了一个简单的递归计算的三重分级Khovanov-Rozansky同调的几个无限家庭的结和链接，包括$(n,nm\pm 1)$和$(n,nm)$环面链接$n,m\geq 1$。我们解释我们的结果在加泰罗尼亚组合学，证明一个猜想的Gorsky的。我们的计算结果与Hilbert格式和有理DAHA的预测结果一致，也证明了Gorsky-Oblomkov-Rasmussen-Shende定理在这些情况下的正确性。此外，我们的研究结果提出了一个拓扑解释的对称功能的上下文中出现的$m$ -洗牌猜想的Haglund-Haiman-Loehr-Remmel-Ulyanov。

2017-04-05math.COmath.GTmath.RT

Affine Laumon spaces and a conjecture of Kuznetsov

We prove a conjecture of Kuznetsov stating that the equivariant K-theory of affine Laumon spaces is the universal Verma module for the quantum affine algebra U_q(gl_n^). We do so by reinterpreting the action of the quantum toroidal algebra U_q(gl_n^^) on the K-theory from [14] in terms of the shuffle algebra studied in [12], which constructs an embedding of U_q(gl_n^) into U_q(gl_n^^)

2020-08-21math.AGmath.RT