Generalized cohomology for irreducible tensor fields of mixed Young symmetry type

E型量子仿射代数1级不可约最高权与Fock空间晶体的Young wall实现

Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

本文为$E_{6}^{(1)}$、$E_{7}^{(1)}$和$E_{8}^{(1)}$型量子仿射代数的1级不可约最高权与Fock空间表示晶体基构造Young wall模型，通过彩色块的堆叠与增减描述晶体结构。

Tensor fields of mixed Young symmetry type and N-complexes

We construct $N$-complexes of non completely antisymmetric irreducible tensor fields on $\mathbb R^D$ which generalize the usual complex $(N=2)$ of differential forms. Although, for $N\geq 3$, the generalized cohomology of these $N$-complexes is non trivial, we prove a generalization of the Poincaré lemma. To that end we use a technique reminiscent of the Green ansatz for parastatistics. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma. We furthermore identify the nontrivial part of the generalized cohomology. Many of the re…

Higher level affine crystals and Young walls

Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals $\mathcal{B}(λ)$ for quantum affine algebras of type $A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}$. The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the constructions of crystal graphs.

复形簇的局部相交上同调

Local intersection cohomology of varieties of complexes

该研究通过Lusztig量子群几何方法与典范基元素显式构造，实现了复形簇不可约分支局部相交上同调的计算。

On irreducibility of tensor products of Yangian modules associated with skew Young diagrams

We study the tensor product $W$ of any number of "elementary" irreducible modules $V_1,...,V_k$ over the Yangian of the general linear Lie algebra. Each of these modules is determined by a skew Young diagram and a complex parameter. For any indices $i,j=1,...,k$ there is a canonical non-zero intertwining operator $A_{ij}$ between the tensor products $V_i\otimes V_j$ and $V_j\otimes V_i$. This operator is defined up to a scalar multipler. We show that the tensor product $W$ is irreducible, if and only if all operators $A_{ij}$ with $i<j$ are invertible. This implies that the Yangian module $W$ …

Affine crystals of type $D_n^{(1)}$ and Young walls

We give a new realization of arbitrary level perfect crystals and arbitrary level irreducible highest weight crystals of type $D_n^{(1)}$, in the language of Young walls. The notions of splitting of blocks and slices play crucial roles in the construction of crystals. The perfect crystals are realized as the set of equivalence classes of slices. And the irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls which, in turn, are concatenations of slices.