The tensor product crystal $B_1\otimes B_2$ and its two-case convention

Let $B_1$ and $B_2$ be crystals, meaning the data $(B,wt,{\epsilon }_i,{\phi }_i,{\tilde{e}}_i,{\tilde{f}}_i)$ attached to a crystal basis $(L,B)$ of an integrable module by The crystal graph, ${\epsilon }_i$, ${\phi }_i$ and the weight function and Crystal lattices, crystal bases and their crystals: $wt:B\to P$ is the weight function, ${\epsilon }_i(b)=\max \{n:{\tilde{e}}_i^{n}b\mathrm{\ne }0\}$ and ${\phi }_i(b)=\max \{n:{\tilde{f}}_i^{n}b\mathrm{\ne }0\}$ are finite, ${\phi }_i(b)={\epsilon }_i(b)+⟨h_i,wt(b)⟩$, and ${\tilde{e}}_i,{\tilde{f}}_i$ are the partial maps $B\to B\cup \{0\}$ of The Kashiwara operators ${\tilde{e}}_i$ and ${\tilde{f}}_i$ of an integrable module.

The tensor product crystal $B_1\otimes B_2$ is the following data on $B_1\times B_2$, whose elements are written $b_1\otimes b_2$:

(i) $wt(b_1\otimes b_2):=wt(b_1)+wt(b_2)$;

(ii) ${\tilde{f}}_i(b_1\otimes b_2):={\tilde{f}}_i{b}_1\otimes b_2$ if ${\phi }_i(b_1)>{\epsilon }_i(b_2)$, and ${\tilde{f}}_i(b_1\otimes b_2):=b_1\otimes {\tilde{f}}_i{b}_2$ if ${\phi }_i(b_1)\le {\epsilon }_i(b_2)$;

(iii) ${\tilde{e}}_i(b_1\otimes b_2):={\tilde{e}}_i{b}_1\otimes b_2$ if ${\phi }_i(b_1)\ge {\epsilon }_i(b_2)$, and ${\tilde{e}}_i(b_1\otimes b_2):=b_1\otimes {\tilde{e}}_i{b}_2$ if ${\phi }_i(b_1)<{\epsilon }_i(b_2)$.

By convention a product is $0$ as soon as the operator applied to the selected factor is $0$; the left factor is preferred exactly in the displayed cases, so the assignment is left-to-right, matching the left-to-right order of the module tensor product $M_1\otimes M_2$ under the coproduct of The Drinfeld–Jimbo formulas define a Hopf algebra. The equality case ${\phi }_i(b_1)={\epsilon }_i(b_2)$ is split: ${\tilde{e}}_i$ acts on the first factor while ${\tilde{f}}_i$ acts on the second. The asymmetry is not free: with the same strict or non-strict comparison used for both operators the partial-inverse axiom fails, as the counterexample on the companion page records, while the displayed convention is internally consistent by the lemma on this page computing ${\epsilon }_i,{\phi }_i$ of $B_1\otimes B_2$.

The functions ${\epsilon }_i,{\phi }_i$ of $B_1\otimes B_2$ are read off from the two-case operators as maximal iterated powers, exactly as in The crystal graph, ${\epsilon }_i$, ${\phi }_i$ and the weight function. This definition only fixes combinatorial data and asserts nothing about an arbitrary pair of crystals: it does not claim that every such datum arises from a module tensor product. When $B_1,B_2$ are the crystals of crystal bases of integrable modules, the tensor product theorem later on this page proves that $(L_1{\otimes }_{A_0}{L}_2,B_1\otimes B_2)$ is a crystal basis of $M_1\otimes M_2$ with these operators, and only through that theorem are the values finite and nonnegative a priori. No choice principle is used.