Double leaves form graded $R$-bases of type-A diagrammatic Hom spaces

Given: The category $D$ with its presentation (The type-A diagrammatic Soergel category and its candidate bimodule functor), whose candidate assignment on generators respects the relations by (The type-A diagrammatic relations hold for Soergel bimodules), and words $\underset{‾}{x},\underset{‾}{y}$ as above.

$D$ is the $\mathbb{Q}$-linear monoidal category generated by colored strands, polynomial boxes, dots, trivalent vertices and the $2m_{st}$-valent vertices, with the degrees of Elias–Williamson, Definition 5.1 and the relations of §§5.1–5.6, so that $D$ is graded and its generators have those degrees (The type-A diagrammatic Soergel category and its candidate bimodule functor).

The elements of $S_n$ and the subexpression relation come from the Coxeter presentation of Type-A reduced words and the Coxeter presentation: a subexpression of $\underset{‾}{x}$ is a word obtained by replacing each letter $x_a$ by either $x_a$ (a step of type $1$) or $1$ (a step of type $0$), and each step is moreover of "up" or "down" type according to whether the length of the partial product increases or decreases; writing $U_0,U_1,D_0,D_1$ for the four possibilities, the defect is $d(e):=\mathrm{\#}{U}_0-\mathrm{\#}{D}_0$, an integer, following Elias–Williamson §2.4.

Imported construction (Elias–Williamson, §6.1, Construction 6.1 with Figure 2): for every $(\underset{‾}{x},e)$ expressing $w$ and every $k\le \mathrm{\ell }(\underset{‾}{x})$ the truncated light leaf on the first $k$ letters is obtained from the truncated light leaf on the first $k-1$ letters by a rex move $\beta$ when the $k$-th letter is a down step, followed by exactly one of four local moves, namely the dot $U_0$ of degree $+1$, the identity $U_1$ of degree $0$, the merging trivalent vertex $D_0$ of degree $-1$ and the cap $D_1$ of degree $0$; the recursion is thus on the prefixes of $\underset{‾}{x}$, each local move has degree equal to the change in the defect, so the resulting light leaf has degree $d(e)$ (The type-A diagrammatic Soergel category and its candidate bimodule functor).

Imported statement (Elias–Williamson, Theorem 6.11, with Proposition 6.12 and Corollary 6.13): the double leaves form a free $R$-basis of ${\mathrm{Hom}}_D(B_{\underset{‾}{x}},B_{\underset{‾}{y}})$, the light leaves ${\mathrm{L}\mathrm{L}}_{\underset{‾}{x},e}$ with $w=e$ form a free $R$-basis of ${\mathrm{Hom}}_D(B_{\underset{‾}{x}},\mathbf{1})$, and all hom spaces of $D$ are free graded $R$-modules (The type-A diagrammatic Soergel category and its candidate bimodule functor).

Imported input (Elias–Williamson, §7, Proposition 7.6 with the coupled maxwidth induction of §7 applied to the lower-term ideals): with the lower-term ideals $I_w$ generated by the maps killing the top summands, every morphism of $D$ between Bott–Samelson objects lies in the span of the double leaves modulo the ideal of strictly lower terms, which supplies the spanning half of [F4]; the proof uses only the relations of $D$ and the induction on the maximum width of a graph (The type-A diagrammatic Soergel category and its candidate bimodule functor).

Imported input (Elias–Williamson, Proposition 6.9, with the localization argument of §6.3): after localization the space of maps between Bott–Samelson objects is a direct sum of terms indexed by pairs of subsequences, these terms carry a partial order, and the double-leaf maps are upper triangular with respect to it with an invertible diagonal; hence the double leaves are linearly independent over the fraction field of $R$ (The type-A diagrammatic Soergel category and its candidate bimodule functor).