Split Grothendieck rings of the type-A Soergel categories

Setting. Fix $n\ge 2$ and the two graded additive monoidal categories of this page: the type-A Soergel category ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$, the idempotent completion of the category of Bott–Samelson bimodules (The type-A Soergel category ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$), and the Karoubi envelope $\mathrm{Kar}(D)$ of the type-A diagrammatic category $D$ (The type-A diagrammatic Soergel category and its candidate bimodule functor). Both are graded: an object $X$ has a shift $X(1)=X\{-1\}$, and both are monoidal with unit the one-object bimodule $R$ respectively the empty word. Both are additive in the sense of Additive category.

The split Grothendieck group. For such a category $C$ choose a small skeleton, that is, a set $\mathrm{sk}(C)$ of representative objects meeting every isomorphism class exactly once — for ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ one may take the graded direct summands of finite direct sums of shifts of Bott–Samelson bimodules with a fixed underlying set of words, and for $\mathrm{Kar}(D)$ the idempotents in finite direct sums of shifts of words, so that the representatives form a set rather than a proper class. Define $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}(C):=({⨁}_{X\in \mathrm{sk}(C)}\mathbb{Z}[X])/⟨[X]-[X^{\mathrm{\prime }}]-[X^{\mathrm{\prime }\mathrm{\prime }}]:X\cong X^{\mathrm{\prime }}\oplus X^{\mathrm{\prime }\mathrm{\prime }}\text{ in }C⟩,$ the free abelian group on the isomorphism classes of representative objects modulo the relations $[X]=[X^{\mathrm{\prime }}]+[X^{\mathrm{\prime }\mathrm{\prime }}]$ whenever $X$ is isomorphic to a direct sum of $X^{\mathrm{\prime }}$ and $X^{\mathrm{\prime }\mathrm{\prime }}$.

Ring structure. For representatives $X,Y$ the tensor product $X\otimes Y$ is again an object of $C$, whose isomorphism class is represented by a unique element of the skeleton; the assignment $(X,Y)\mapsto X\otimes Y$ is additive in each variable up to isomorphism, so $[X][Y]:=[X\otimes Y]$ descends to a well-defined bilinear product on $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}(C)$, with the unit $1:=[\mathbf{1}]$ the class of the monoidal unit $R$ (respectively the empty word). Associativity and unitality of the tensor product make $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}(C)$ a ring; it is not commutative in general, and $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}({\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n)$ is not commutative for $n\ge 3$, since the classes of two adjacent one-color products do not commute. The grading shift is an additive autoequivalence commuting with the tensor product, so setting $v[X]:=[X(1)]=[X\{-1\}]$ equips $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}(C)$ with the structure of a $\mathbb{Z}[v,v^{-1}]$-algebra, with $v^{-1}[X]=[X(-1)]=[X\{1\}]$.

The two charts. For the bimodule category the classes are those of the Bott–Samelson products, $[B_{i_1}{\otimes }_R\cdots {\otimes }_R{B}_{i_r}]$, and for every object $X$ the shift rule is $[X\{k\}]=v^{-k}[X]$; for the diagrammatic category the class of a word is $[B_{\underset{‾}{i}}]$, with the same shift rule. The standard bimodules $R_x$, ${\mathrm{\Delta }}_x(d)$ and ${\mathrm{\nabla }}_x(d)$ with $x\mathrm{\ne }e$ are not objects of ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ — they occur only as the successive quotients of the support flags of its objects — so this definition attaches no object class to them; the Hecke algebra's basis elements indexed by $x$ are written $T_x,{\tilde{T}}_x,H_x$, as in The type-A Hecke algebra in Soergel normalization. Under the dictionary of The type-A diagrammatic Soergel category and its candidate bimodule functor the two rings are compared by sending $[B_{\underset{‾}{i}}]$ to $[B_{i_1}{\otimes }_R\cdots {\otimes }_R{B}_{i_r}]$; the comparison is an isomorphism only after the equivalence of the two categories is proved later on this page.

(a) Local scope. This definition supplies only the split rings of the two categories used on this page, on explicitly chosen small skeletons; it makes no claim about general Grothendieck groups, Cartan pairings, or the Grothendieck groups of arbitrary additive or abelian categories, which belong to the general homological-algebra track. The only group-theoretic input is that a set of representatives exists, and the only ring-theoretic input is bifunctoriality and additivity of $\otimes$, both of which hold for the two categories named here.

(b) The relation with the internal shift. In the library convention $M\{r{\}}_d=M_{d-r}$ the internal shift by $r$ moves every generator degree up by $r$, so $[X\{r\}]=v^{-r}[X]$: the element $v$ of the Laurent ring is the class of the shift that lowers degrees, which is the Elias–Williamson shift $(1)$ and the shift appearing in $B_i=R{\otimes }_{R^{s_i}}R(1)$. The one-color square $B_i{\otimes }_R{B}_i\cong B_i(1)\oplus B_i(-1)$ of The rank-one Soergel bimodule square splits therefore reads $[B_i{]}^2=(v+v^{-1})[B_i]$.

(c) Products of generators. The product rule gives $[B_{\underset{‾}{i}}]=[B_{i_1}]\cdots [B_{i_r}]$, and for $\mathrm{\mid }i-j\mathrm{\mid }>1$ the interchange isomorphism of Distant Soergel generators commute gives $[B_i][B_j]=[B_j][B_i]$ in $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}$. No generation statement is made here: the skeleton of ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ also contains idempotent summands of words, and in a split Grothendieck ring the defining relations give only $[X]=[Y]-[Z]$ for $Y\cong X\oplus Z$, so the class of such a summand is not in general determined by the classes of the ambient words. Generation by the classes of the $B_i$ together with $v^{\pm 1}$ is therefore established later on this page, from the character isomorphism, and is not an input to it.

(d) Small and empty cases. For $n\le 1$ there are no simple reflections, so the only word is the empty one and every object of ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$ is a finite direct sum of shifts of the unit $R$, whence $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}=\mathbb{Z}[v,v^{-1}]\cdot [R]$; for $n=2$ the only words are powers of $s_1$, the one-color square exhibits each $B_1^{\otimes k}$ with $k\ge 1$ as a finite direct sum of shifts of $B_1$, and $[R]$ is the unit. In every case the empty-direct-sum relation $[0]=0$ is the case $X\cong X\oplus 0$ of the defining relations, and no choice principle is used: each skeleton is a set of concrete objects, and $\mathbb{Z}[X]$ is the free group on that set.