Highest weight elements, lowering-generated sets and highest weight crystals

Let $B$ be a crystal, meaning the data $(B,wt,{\epsilon }_i,{\phi }_i,{\tilde{e}}_i,{\tilde{f}}_i)$ attached to a crystal basis or to a tensor product of such data (The crystal graph, ${\epsilon }_i$, ${\phi }_i$ and the weight function, The tensor product crystal $B_1\otimes B_2$ and its two-case convention).

A vertex $b\in B$ is a highest weight element (a highest vertex, a maximal vector) of weight $\lambda \in P$ if $wt(b)=\lambda$ and ${\tilde{e}}_{i}b=0$ for every $i\in I$. The lowering-generated set of $b$ is

$F(b):=\{{\tilde{f}}_{i_1}\cdots {\tilde{f}}_{i_r}b:r\ge 0,i_1,\dots ,i_r\in I\}\setminus \{0\},$

the set of all nonzero iterated lowering images of $b$; it contains $b$ and is closed under nonzero images of every ${\tilde{f}}_i$. It need not be closed under raising: at the bottom $b$ of a two-vertex rank-one string, $F(b)=\{b\}$ while ${\tilde{e}}_{i}b$ is the top vertex. The connected component $C(b)$ of $b$ is the smallest subset of $B$ containing $b$ and closed under nonzero images of ${\tilde{e}}_i$ and ${\tilde{f}}_i$ for all $i$; by the partial-inverse axiom this is the connected component of the underlying coloured graph.

$B$ is a highest weight crystal of highest weight $\lambda$ if $B$ has a highest weight element $b$ of weight $\lambda$ and $B=F(b)$.

The definition deliberately separates two conditions. Always $F(b)\subseteq C(b)$, and a highest vertex need not generate $B$: in a tensor product $B_1\otimes B_2$ a vertex with ${\epsilon }_i(b_1)=0$ and ${\epsilon }_i(b_2)\le {\phi }_i(b_1)$ for all $i$ is highest while $B_1\otimes B_2$ is in general larger than its lowering-generated set (the companion page computes both cases). Conversely, for abstract crystals the two conditions do not determine $B$ up to isomorphism: the source literature records that not every connected crystal with a highest weight element of weight $\lambda$ is isomorphic to $B(\lambda )$. In this library crystals come from crystal bases of modules, and uniqueness for those is a theorem proved later on this page, not a consequence of this definition.

For the canonical objects, $B(\lambda )=\mathcal{B}(\lambda )$ denotes the crystal of the simple highest weight module $V(\lambda )=L(\lambda )$ (The Kashiwara lattices $\mathcal{L}(\lambda )$ and $\mathcal{L}(\mathrm{\infty })$ and their generating sets) and $P^{+}$ the dominant integral labels (Kac moody integral and dominant integral weights); that $B(\lambda )$ is a highest weight crystal of highest weight $\lambda$ is the next theorem. The operators ${\tilde{e}}_i,{\tilde{f}}_i$ used throughout are those of The Kashiwara operators ${\tilde{e}}_i$ and ${\tilde{f}}_i$ of an integrable module. No choice principle is used.