Abstract |
We study the highest weight crystals of type $D$ in a combinatorial viewpoint. In this thesis, we study mainly the crystals of the negative half of the quantum group $U_q(D_n)$ and an integrable highest weight irreducible $U_q(D_n)$-module. This admits several applications such as a new combinatorial model of Kirillov--Reshetikhin crystals $B^{n, s}$ of untwisted affine type $D$ related to the spin node, a new formula for the branching rule from ${GL}_n$ to ${O}_n$ and a crystal theoretic interpretation of Robinson--Schensted--Knuth type correspondence for type $D$. |