Francois Bergeron, Francois Lamontagne
The purpose of this paper is to give an explicit description of the irreducible decomposition of the multigraded S_n-module of coinvariants of S_n x S_n. Many of the results presented can be extended to analogous questions for other finite reflection group.
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We describe a general conjecture on how one may derive from the generic bosonic case all structural properties of multivariate diagonal coinvariant modules in $k$ sets of $n$ commuting variables (bosons), and $j$ sets of $n$ anticommuting variables (fermions).
We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group $W=G(m,p,n)$ and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category $\cO$ for the rational Cherednik algebra of $W$.
We explain how to use representation theory to give a lower bound on the dimension of the quotient ring by type $B_n$ diagonal invariants that improves upon the current known lower bound $(2n+1)^n$ by a quadratic polynomial in $n$.
For finite complex reflexion groups, we consider the graded $W$-modules of diagonally harmonic polynomials in $r$ sets of variables, and show that associated Hilbert series may be described in a global manner, independent of the value of $r$.
The diagonal coinvariant ring of a complex reflection group
本研究针对不可约复反射群，构造了Cherednik代数的一个高维不可约表示，提出其为与对角不变量环相关的最大表示，并揭示了对称群乘积点作用在构造中的关键作用。
Let $Θ_n = (θ_1, \dots, θ_n)$ and $Ξ_n = (ξ_1, \dots, ξ_n)$ be two lists of $n$ variables and consider the diagonal action of $\mathfrak{S}_n$ on the exterior algebra $\wedge \{ Θ_n, Ξ_n \}$ generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ Θ_n, Ξ_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. In joint work with Rhoades we gave a basis for the maximal degree components of this ring where the action of $\mathfrak{S}_n$ could be interpreted combinatorially via n…
Diagonal supersymmetry for coinvariant rings
本文证明有限群G的玻色-费米余不变环具有自然的U(gl(k|j))⊗C[G]-模结构，其级数可分解为超Schur函数与G的不可约特征之积，系数与k,j无关，并由此证明了F. Bergeron的“对角超对称性”猜想。
Cluster parking functions II: $q,t$-dihedral sieving via diagonal coinvariants
研究簇停车函数复形，考虑其底层二面体对称性，在二面体群与对称群乘积作用下，猜想其约化同调与对角余不变量同构，主要贡献是定义二面体自同构群并讨论相关性质。
Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = \mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $\mathfrak{S}_n$-modules. We relate the Hilbert series of…
We calculate the Borel-Moore homology of affine Springer fibers of type $A$ associated to some regular semisimple nil elliptic elements. As a result, we obtain bigraded $\mf{S}_{n}$-modules whose bigraded Frobenius series are generalization of the symmetric functions introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov.
The purpose of this paper is to give an explicit description of the trivial and alternating components of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups.
Universal series for dihedral group coinvariant rings
本文利用对角超对称性，将二面体群对角不变环的基与级数推广至任意玻色和费米变量，确定了通用多重分次特征级数、Hilbert级数和单项式基。
Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly-graded S_n module can be expressed using the Frobenius characteristic map as \nabla e_n, where e_n is the n-th elementary symmetric function, and \nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for \nabla e_n and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. Th…
Sign components of diagonal superspace coinvariants
本文证明了对角超空间余不变环的符号分量同构，其重数由薛定谔多项式给出，解决了两个猜想，并加强了环面结同调与符号分量的联系。
This is the first in a series of papers in which we describe explicit structural properties of spaces of diagonal rectangular harmonic polynomials in $k$ sets of $n$ variables, both as $GL_k$-modules and $S_n$-modules, as well as some of there relations to areas such as Algebraic Combinatorics, Representation Theory, Algebraic Geometry, Knot Theory, and Theoretical Physics. Our global aim is to develop a unifying point of view for several areas of research of the last two decades having to do with Macdonald Polynomials Operator Theory, Diagonal Coinvariant Spaces, Rectangular-Catalan Combinato…
We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of N analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
Higher Specht bases for generalizations of the coinvariant ring
本文为广义余不变环 $R_{n,k}$ 和Garsia-Procesi模 $R_\mu$ 的推广构造了更高Specht基，并证明了两行分区情形下的猜想，进而为Griffin定义的 $R_{n,k,\mu}$ 无限子族提供了基。
The Hilbert series of the superspace coinvariant ring
本文研究了超空间余不变环 $SR_n$，计算了其双重分次 Hilbert 级数，证明了刻画相伴调和空间的算子定理，并证实了多位学者的猜想。
In the context of the ring Q[x,y], of polynomials in 2n variables x=x1,...,x_n and y=y1,...,yn, we introduce the notion of diagonally quasi-symmetric polynomials. These, also called "diagonal Temperley-Lieb invariants", make possible the further introduction of the space of "diagonal Temperley-Lieb harmonics" and "diagonal Temperley-Lieb coinvariant space". We present new results and conjectures concerning these spaces, as well as the space obtained as the quotient of the ring of diagonal Temperley-Lieb invariants by the ideal generated by constant term free diagonally symmetric invariants. We…
Regularity and stable ranges of FI-modules
该研究给出FI-模正则性及多种稳定化形式下稳定范围的精细紧界，应用于对角不变代数得到显式稳定范围，并改进了流形有序配置空间、一般线性群同余子群的相关稳定范围。
Higher Specht polynomials under the diagonal action
本文引入对角作用下两组变量的高阶 Specht 多项式，推广经典构造，并给出钩形 Garsia--Haiman 模的高阶 Specht 基，导出其双次数 Frobenius 级数的新公式。
Given an $n\times n$ array $M$ ($n\ge 7$), where each cell is colored in one of two colors, we give a necessary and sufficient condition for the existence of a partition of $M$ into $n$ diagonals, each containing at least one cell of each color. As a consequence, it follows that if each color appears in at least $2n-1$ cells, then such a partition exists. The proof uses results on completion of partial Latin squares.
Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence it is proved the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproduct are a pure combinatorial algorithm to describe the irreducible decomposition of the tensor product of two irreducible representations of the symmetric group, and new symmetry results on permutation enumeration with respect to descent sets.
Young-lattice diagonals and a doubly graded multiple-zeta decomposition of $e^γ$
本研究引入双变量生成函数与系数数组，建立Young格对角分拆与e^γ的Abel正则化多重zeta展开的双分次对应，给出钩型与非钩型贡献的规范分解及渐近结果。
Star Decompositions of a Cyclic Polygon
本文研究了循环多边形的最大星形分解的性质，证明了两个最大分解可通过对角线翻转转换，并给出了最大分解中对角线数量的计算公式。
We study the combinatorial description of the LDU decomposition of totally positive matrices. We give a description of the lower triangular L, the diagonal D, and the upper triangular U matrices of the LDU decomposition of totally positive matrices in terms of the combinatorial structure of essential planar networks described by Zelvinsky and Fomin. Similarly, we find a combinatorial description of the inverses of these matrices. In addition, we provide recursive formulae for computing the L, D, and U matrices of a totally positive matrix.
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