Peter Tingley
Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal for affine sl(n), where the vertices are indexed by certain partitions. He showed that special cases of this construction agree with the Misra-Miwa realization and with Berg's ladder crystal. Here we show that another special case is naturally isomorphic to a realization using Nakajima's monomial crystal.
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An explicit description of the virtualization map for the (modified) Nakajima monomial model for crystals is given. We give an explicit description of the Lusztig data for modified Nakajima monomials in type $A_n$.
In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, w…
The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\mathfrak{g}$, and depends on a collection of parameters $\mathbf{R}$. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar. In type $A$, we show the product monomial crystal is the cry…
Decomposition Theorem for Product of Fundamental Crystals in Monomial Realization of type $C_n$
本文研究$C_n$型基础晶体在单项式实现下的洛朗单项式自然乘积，证明其仍具晶体结构并给出不可约晶体分解规则，指出该规则与基础晶体张量积的分解规则通常存在差异。
A conjecture on monomial realizations and polyhedral realizations for crystal bases
本文提出并部分证明了一个猜想，该猜想建立了可对称化 Kac-Moody 代数晶体基的多面体锥显式形式与单项式实现之间的联系，并在经典李代数、秩2 Kac-Moody 代数及经典仿射李代数情形下验证了其正确性。
We give a realization of the Kirillov--Reshetikhin crystal $B^{1,s}$ using Nakajima monomials for $\widehat{\mathfrak{sl}}_n$ using the crystal structure given by Kashiwara. We describe the tensor product $\bigotimes_{i=1}^N B^{1,s_i}$ in terms of a shift of indices, allowing us to recover the Kyoto path model. Additionally, we give a model for the KR crystals $B^{r,1}$ using Nakajima monomials.
The weight function for monomial crystals of affine type
本文针对仿射型 $A_n^{(1)}$ 与 $B_3^{(1)}$ 的 Nakajima 单项式晶体，推导并给出其仿射权的表达式。
We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes p dividing the number, M(a,b,c), of plane partitions contained inside an arbitrary box of given sides a,b,c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a,b,c) fails to have the WLP in chara…
On the nonnegativity of monomial immanants for hook partitions
本文解决了 Stembridge 提出的关于单项式隐式量非负性的开放问题，证明了对于所有钩子分区，当矩阵的所有低阶子式非负且分区长度受限时，隐式量非负。
We consider the homogeneous components U_r of the map on R = k[x,y,z]/(x^A, y^B, z^C) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of submatrices of an arbitrary Toeplitz matrix using Schur polynomials, and use this to give a relationship between Smith normal form entries of U_r. We also give a bijective proof of an identity proven by J. Li and F. Zanello equating the determinant of the middle homogeneous component U_r when (A, B, C) = (a + b, a + c, b + c) to the number of plane partitions in an a by b by c box. Finally, we prove that, for certain vecto…
We describe an explicit crystal morphism between Nakajima monomials and monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C. This morphism provides a connection between arbitrary Nakajima monomials and Nakashima Kashiwara tableaux. This yields a translation of Nakajima monomials to the Littelmann path model. Furthermore, as an application of our results we describe an insertion scheme for Nakajima monomials compatible to the insertion scheme for tableaux.
The purpose of this paper is to find a new way to prove the $n!$ conjecture for particular partitions. The idea is to construct a monomial and explicit basis for the space $M_μ$. We succeed completely for hook-shaped partitions, i.e., $μ=(K+1,1^L)$. We are able to exhibit a basis and to verify that its cardinality is indeed $n!$, that it is linearly independent and that it spans $M_μ$. We derive from this study an explicit and simple basis for $I_μ$, the annihilator ideal of $Δ_μ$. This method is also successful for giving directly a basis for the homogeneous subspace of $M_μ$ consisting of el…
For each $n\geqslant3$, we construct an uncountable family of models of the crystal of the basic $U_q(\hat{\mathfrak{sl}}_n)$-module. These models are all based on partitions, and include the usual $n$-regular and $n$-restricted models, as well as Berg's ladder crystal, as special cases.
We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight. Cylindric plane partitions actually parametrize a basis for the tensor product of an irreducible representation with the space spanned by all partitions. We use this to calculate the partition function for a system of random cylindric plane partitions. We also observe a form of rank level duality. Fi…
A $q$-deformation of enriched $P$-partitions (extended abstract)
本文提出增广 $P$-分拆的 $q$-形变框架，构造在 Gessel 基本函数与 Stembridge 峰函数间插值的拟对称函数族及其单项式基。
For a polynomial f: {-1, 1}^n --> C, we define the partition function as the average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is a parameter. We present a quasi-polynomial algorithm, which, given such f, lambda and epsilon >0 approximates the partition function within a relative error of epsilon in N^{O(ln n -ln epsilon)} time provided |lambda| < 1/(2 L sqrt{deg f}), where L=L(f) is a parameter bounding the Lipschitz constant of f from above and N is the number of monomials in f. As a corollary, we obtain a quasi-polynomial algorithm, which, given such an f with coe…
We introduce a family of ideals $I_{n,λ,s}$ in $\mathbb{Q}[x_1,\dots,x_n]$ for $λ$ a partition of $k\leq n$ and an integer $s \geq \ell(λ)$. This family contains both the Tanisaki ideals $I_λ$ and the ideals $I_{n,k}$ of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings $R_{n,λ,s}$ as symmetric group modules. When $n=k$ and $s$ is arbitrary, we recover the Garsia-Procesi modules, and when $λ=(1^k)$ and $s=k$, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for $R_{n,λ,s}$, unifying the monomial bases stu…
In a recent work, Keith and Xiong gave a refinement of Glaisher's theorem by using a Sylvester-style bijection. In this paper, we introduce two families of colored partitions, flat and regular partitions, and generalize the bijection of Keith and Xiong to these partitions. We then state two results, the first at degree one, where partitions have parts with primary colors, and the second result at degree two for secondary-colored partitions, using the result of the first paper of this series. These results allow us to easily retrieve the Frenkel-Kac character formulas of level one standard modu…
We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of…
Partition identities from higher level crystals of $A_1^{(1)}$
本文利用多重基分拆理论研究了仿射李代数 $A_1^{(1)}$ 各层标准模的完美晶体，证明了一族带有绝对值差条件、与 Andrews-Gordon 及 Meurman-Primc 恒等式相关的分拆恒等式。
Schur partition theorems via perfect crystal
受对称群自旋模表示启发，提出两类 Schur 正则分拆的推广，证明其具有基本晶体结构，并重新证明 Schur 定理及给出 Andrews 猜想的无计算机证明。
Decomposition of Polysymmetric Functions and Stack Partitions
本文通过堆栈划分研究多项对称函数，提出两种新的带符号基并给出转换公式，扩展了对称函数理论的恒等式到多项对称环境。
PBW bases and marginally large tableaux in type D
本文刻画D型$B(\infty)$的边缘大表实现与基于最长词良好约化表达式的PBW单项式实现之间的唯一晶体同构。
Cyclic Sieving for Staircase Plane Partitions via Crystals and Electrical Networks
研究高度为二的阶梯平面划分提升作用的循环筛选结果，利用自旋晶体张量幂解释及电网络空间坐标环二次部分的灌木丛基证明，还探讨了所有度数上电典范基存在时对所有高度阶梯平面划分提升的循环筛选意义。
Vazirani and the author \cite{BV} gave a new interpretation of what we called $\ell$-partitions, also known as $(\ell,0)$-Carter partitions. The primary interpretation of such a partition $λ$ is that it corresponds to a Specht module $S^λ$ which remains irreducible over the finite Hecke algebra $H_n(q)$ when $q$ is specialized to a primitive $\ell^{th}$ root of unity. To accomplish this we relied heavily on the description of such a partition in terms of its hook lengths, a condition provided by James and Mathas. In this paper, I use a new description of the crystal $reg_\ell$ which helps exte…
Consider Kashiwara's crystal associated to a highest weight representation of a symmetric Kac-Moody algebra. There is a geometric realization of this object using Nakajima's quiver varieties, but in many particular cases it can also be realized by elementary combinatorial methods. Here we propose a framework for extracting combinatorial realizations from the geometric picture: We construct certain torus actions on the quiver varieties and use Morse theory to index the irreducible components by connected components of the subvariety of torus fixed points. We then discuss the case of affine sl(n…
A remark on certain restricted plane partitions and crystal melting model
本文利用Okounkov、Reshetikhin和Vafa提出的晶体熔化模型，给出了两类平面分拆配分函数的计算公式，并推导出极限边界下平面分拆配分函数的乘积公式，证明了该配分函数与双P^1模型的开弦振幅等价，同时得到了z轴方向极限边界下对称平面分拆配分函数的乘积公式。
Demazure crystals for flagged set-valued reverse plane partitions
本文在带旗集值反向平面分割上构建Demazure晶体，统一推广了已知结构，并由此得到混合Grothendieck多项式的key多项式展开，进而将其表示为稳定与对偶稳定Grothendieck多项式的组合。
In this paper we give an alternate combinatorial description of the "$(\ell,0)$-JM partitions" (see \cite{F}) that are also $\ell$-regular. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas (\cite{JM}). The condition of being an $(\ell,0)$-JM partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-regular partition is that it indicates the irreducibility of the corresponding specialized Specht module over the finite Hecke algebra (see \cite{JM}). We use…
Demazure crystal structure for flagged reverse plane partitions
本文研究带标志反向平面分拆，证明其集合可分解为 Demazure 晶体的不交并，并据此推出带标志对偶稳定 Grothendieck 多项式具有 key 正性。
Crystal Melting, Triality and Partition Functions for Toric Calabi-Yau Fourfolds
本文研究了托里切利卡拉比-尤康四维的晶熔模型，开发了基于周期性四元组的算法，分析了晶体在三重性下的行为，并提出了稳定变量的概念以稳定配分函数。
The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a partition $λ$ is irreducible. This is done by extending the results of James and Mathas. These descriptions depend on the crystal of the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to determine which irreducible modules have a realization as a Specht module. To do this, a new condition of irr…
In the first paper of this series, we gave infinite families of coloured partition identities which generalise Primc's and Capparelli's classical identities. In this second paper, we study the representation theoretic consequences of our combinatorial results. First, we show that the difference conditions we defined in our $n^2$-coloured generalisation of Primc's identity, which have a very simple expression, are actually the energy function with values in $\{0,1,2\}$ for the perfect crystal of the tensor product of the vector representation and its dual in $A_{n-1}^{(1)}$. Then we introduce a…
We refine and generalise a Rogers-Ramanujan type partition identity arising from crystal base theory. Our proof uses the variant of the method of weighted words recently introduced by the first author.
In this paper, we use an M-theory model to conjecture the refined reminiscence of the OSV formula connecting the refined topological string partition function with the refined BPS states partition function for the toric Calabi-Yau threefolds without any compact four cycles. Further, we show how to use the vertex operators in 2d free fermions to reproduce the refined BPS states partition function for the $\mathbb{C}^3$ case and the wall-crossing formulas of the refined BPS states partition function for the resolved conifold and $\mathcal{O}(-2)\oplus\mathcal{O}\rightarrow \mathbb{P}^1$ cases.
Divisible Arm Lengths, Crystal Reflections, and Enumeration of Newly Found Decomposition Columns
本文研究并枚举了满足特定钩长可除条件的划分，证明了晶体反射保持d-平衡性质，并给出了e-正则划分在块中的计数公式。
The nonsingular variety G_T parametrizes all graded ideals I of R=k[x,y] for which the Hilbert function H(R/I)=T. The variety G_T has a natural cellular decomposition: each cell V(E) corresponds to a monomial ideal E for which H(R/E)=T. Given a monomial ideal E, the quotient R/E has a basis of monomials in the shape of a partition P(E), having "diagonal lengths" T. The dimension of the cell V(E) is the number of difference one hooks of P(E). We show that G_T is birational to a certain product SGrass(T) of small Grassman varieties, and that over the complexes, the birational map induces an isom…
We describe wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the non-commutative Donaldson-Thomas invariants, given by the modulus square of the topological string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field …
Closed Colored Models and Demazure Crystals
本文提出通过构造可解格子模型及其晶体结构，证明其状态构成Demazure晶体。
Cacti, Toggles, and Reverse Plane Partitions
本文研究 D 型晶体 B(nϖ₁) 上的仙人掌群作用，利用高度 n 的反向平面分拆上的组合切换证明其可由长度一、二的子图元素生成，解决相关猜想。
We survey geometrical and especially combinatorial aspects of generalized Donaldson-Thomas invariants (also called BPS invariants) for toric Calabi-Yau manifolds, emphasizing the role of plane partitions and their generalizations in the recently proposed crystal melting model. We also comment on equivalence with a vicious walker model and the matrix model representation of the partition function.
For integers $e,\ell\geq 2$, the level $\ell$ Fock space has an $\mathfrak{sl}_\infty$-crystal structure arising from the action of a Heisenberg algebra, intertwining the $\widehat{\mathfrak{sl}_e}$-crystal. The vertices of these crystals are charged $\ell$-partitions. We give the combinatorial rule for computing the arrows anywhere in the $\mathfrak{sl}_\infty$-crystal. This allows us to pinpoint the location of any charged $\ell$-partition. As an application, we compute the support of the spherical representation of a cyclotomic rational Cherednik algebra, and in particular, the set of param…
Charging solid partitions
本文研究4维熔融晶体对应的立体分拆，推导其电荷函数，并发现4盒与5盒簇贡献导致相应BPS代数构造更复杂。
Rogers-Ramanujan type identities at $Λ_0$ from perfect crystals of exceptional quantum affine algebras
通过完美晶体推导 exceptional 量子 affine 代数在 $Λ_0$ 处的 Rogers-Ramanujan 类身份，并提供生成函数的系数相等性验证。
Quiver BPS Indices from Crystal Profiles
推导出二维含\(\mathcal{N}\geq2\)超荷的椭圆亏格及其一维、零维对应物的新精确公式，结果表示为对晶体几何/组合结构的离散和，推广了Nekrasov配分函数并揭示新组合学。
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