Standard graph bimodules, support filtrations and characters

Graphs and supports. Keep $k=\mathbb{Q}$, $R=\mathbb{Q}[x_1,\dots ,x_n]$ with $\mathrm{deg}{x}_i=2$, and the affine scheme $V={\mathbb{A}}_k^n=\mathrm{Spec}R$, whose $k$-points are $V(k)=k^n$. Use the place-permuting action of $S_n$ from The standard type-A reflection realization and its polynomial ring. For $w\in S_n$ let $Gr(w):=\mathrm{Graph}(w:V\to V)\subseteq V{\times }_{k}V,Gr(w)(k)=\{(w\lambda ,\lambda ):\lambda \in k^n\},Gr(A):={\bigcup }_{w\in A}Gr(w)(A\subseteq S_n),$ so that $V{\times }_{k}V=\mathrm{Spec}(R{\otimes }_{k}R)$ and $Gr(w)$ is the graph of the map $\lambda \mapsto w\lambda$. A graded $(R,R)$-bimodule $M$ is the same thing as a graded module over $R{\otimes }_{k}R$, and the support of $m\in M$ is the zero set $\mathrm{supp}(m):=V(\mathrm{Ann}(m))\subseteq V\times V$ of its annihilator ideal in $R{\otimes }_{k}R$. For $A\subseteq S_n$ put ${\mathrm{\Gamma }}_{A}M:=\{m\in M:\mathrm{supp}(m)\subseteq Gr(A)\},$ a graded sub-bimodule of $M$. Since $Gr(A)$ is closed with radical ideal $I(Gr(A))={\bigcap }_{w\in A}I(Gr(w))$ (an intersection of prime ideals, since each $Gr(w)$ is a linear subspace of $V\times V$ complementing the second factor), an element $m$ lies in ${\mathrm{\Gamma }}_{A}M$ exactly when $I(Gr(A){)}^{N}m=0$ for some $N\ge 0$: this is the Nullstellensatz together with the Noetherianity of $R{\otimes }_{k}R$. The ideal itself need not annihilate such an element — in $(R{\otimes }_{k}R)\mathrm{/}I(Gr(e){)}^2$ the class of $1$ is supported on $Gr(e)$ but is not killed by $I(Gr(e))$ — which is why the criterion is stated with a power of the ideal. For $i\in \mathbb{Z}$ we write ${\mathrm{\Gamma }}_{\ge i}M:={\mathrm{\Gamma }}_{\{x:\mathrm{\ell }(x)\ge i\}}M,{\mathrm{\Gamma }}_{\le i}M:={\mathrm{\Gamma }}_{\{x:\mathrm{\ell }(x)\le i\}}M,$ using the length function of Type-A reduced words and the Coxeter presentation, while for $y\in S_n$ the $y$-indexed symbols denote the Bruhat principal-set cutoffs of Soergel's Notation 6.1, ${\mathrm{\Gamma }}_{\le y}M:={\mathrm{\Gamma }}_{\{x:x\le y\}}M,{\mathrm{\Gamma }}_{<y}M:={\mathrm{\Gamma }}_{\{x:x<y\}}M,{\mathrm{\Gamma }}_{\ge y}M:={\mathrm{\Gamma }}_{\{x:x\ge y\}}M,{\mathrm{\Gamma }}_{>y}M:={\mathrm{\Gamma }}_{\{x:x>y\}}M,$ with ${\mathrm{\Gamma }}_{y}M:={\mathrm{\Gamma }}_{\{y\}}M$ the elements concentrated on the single graph $Gr(y)$ and ${\mathrm{\Gamma }}_{\mathrm{\ne }y}M:={\mathrm{\Gamma }}_{S_n\setminus \{y\}}M$ the elements concentrated off $Gr(y)$. The integer-indexed symbols of the previous display are the length cutoffs used to define the flags below, and they must not be confused with the element-indexed Bruhat cutoffs: for $y=s_1$ in $S_4$ the submodule ${\mathrm{\Gamma }}_{\le 1}M$ contains $Gr(s_3)$ while ${\mathrm{\Gamma }}_{\le y}M={\mathrm{\Gamma }}_{\{e,s_1\}}M$ does not, and the two cutoffs agree at the extremes, ${\mathrm{\Gamma }}_{\le w_0}={\mathrm{\Gamma }}_{\le \mathrm{\ell }(w_0)}=M$ and ${\mathrm{\Gamma }}_{<e}={\mathrm{\Gamma }}_{\le -1}=0$. So ${\mathrm{\Gamma }}^{y}M:=M\mathrm{/}{\mathrm{\Gamma }}_{\mathrm{\ne }y}M$ is Soergel's $y$-layer quotient; following Soergel's Notation 6.1 we write the two $y$-layer quotients with an underline, ${\underset{‾}{\mathrm{\Gamma }}}_{\le y}M:={\mathrm{\Gamma }}_{\le y}M\mathrm{/}{\mathrm{\Gamma }}_{<y}M,{\underset{‾}{\mathrm{\Gamma }}}_{\ge y}M:={\mathrm{\Gamma }}_{\ge y}M\mathrm{/}{\mathrm{\Gamma }}_{>y}M,$ so that ${\underset{‾}{\mathrm{\Gamma }}}_{\le y}M$ is the quotient of the lower support submodule by ${\mathrm{\Gamma }}_{<y}M$ and ${\underset{‾}{\mathrm{\Gamma }}}_{\ge y}M$ the quotient of the upper support submodule by the part supported strictly above $Gr(y)$.

Standard bimodules. For $w\in S_n$ let $R_w$ be the graded $(R,R)$-bimodule which equals $R$ as a graded $k$-vector space and has $f\cdot x=fx,x\cdot g=w(g)x(f,g,x\in R),$ where $w(g)(\lambda ):=g(w^{-1}\lambda )$; $R_w$ is generated by the element $1$ of degree $0$. Write $(k)$ for the external shift $M(k):=M\{-k\}$ of Associative graded algebras, bimodules, and internal shifts, so that $M(k{)}_d=M_{d+k}$; this is the shift convention of the type-A sources, and it is the one in which $B_i=R{\otimes }_{R^{s_i}}R(1)$ of The Soergel bimodule $B_i$ of a simple reflection has $1\otimes 1$ of degree $-1$. Put ${\mathrm{\Delta }}_w(d):=R_w(d-\mathrm{\ell }(w)),{\mathrm{\nabla }}_w(d):=R_w(d+\mathrm{\ell }(w))(d\in \mathbb{Z}),$ so that ${\mathrm{\Delta }}_w(d)$ is the standard bimodule $R_w$ with its generator in degree $\mathrm{\ell }(w)-d$, and ${\mathrm{\nabla }}_w(d)$ is $R_w$ with its generator in degree $-\mathrm{\ell }(w)-d$. Thus ${\mathrm{\Delta }}_w(0)=R_w\{\mathrm{\ell }(w)\}$, ${\mathrm{\nabla }}_w(0)=R_w\{-\mathrm{\ell }(w)\}$ and ${\mathrm{\Delta }}_e(d)={\mathrm{\nabla }}_e(d)=R(d)$ is generated in degree $-d$; in particular ${\mathrm{\Delta }}_e(0)={\mathrm{\nabla }}_e(0)=R_e=R$.

For graded free $R$-modules in this type-A page, a homogeneous free generator of degree $a$ contributes $v^a$ to graded rank. This is the $v\mapsto v^{-1}$ transform of Soergel's rank convention in his Notation 5.2; all rank formulas below use this convention.

Support filtrations. A $\mathrm{\Delta }$-flag of a graded $R$-bimodule $M$ is a finite chain $0=M_0\subseteq M_1\subseteq \cdots \subseteq M_k=M$ of graded sub-bimodules which refines the length filtration $M={\mathrm{\Gamma }}_{\ge 0}M\supseteq {\mathrm{\Gamma }}_{\ge 1}M\supseteq \cdots \supseteq {\mathrm{\Gamma }}_{\ge \mathrm{\ell }(w_0)+1}M=0$ of $M$ (here ${\mathrm{\Gamma }}_{\ge 0}M=M$ for every bimodule supported on $Gr(S_n)={\bigcup }_{w}Gr(w)$, which all bimodules of this page are, and ${\mathrm{\Gamma }}_{\ge i}M=0$ for $i>\mathrm{\ell }(w_0)$ because no graph $Gr(x)$ with $\mathrm{\ell }(x)>i$ remains) and whose successive quotients are standard bimodules $M_j\mathrm{/}{M}_{j-1}\cong R_{x_j}\{a_j\}$. We require each length layer to be a finite direct sum of shifted graph bimodules of that length, as in Soergel's Definition 5.4. Thus the quotients carry a numbering with $\mathrm{\ell }(x_1)\ge \cdots \ge \mathrm{\ell }(x_k)$ and each $R_{x_j}\{a_j\}$ is one graph, so that $M$ has a "support flag" in the sense of Soergel's $F_{\mathrm{\Delta }}$. A $\mathrm{\nabla }$-flag has the same direct-sum condition on its length layers, with the reverse length filtration ${\mathrm{\Gamma }}_{\le i}M$ and $\mathrm{\ell }(x_1)\le \cdots \le \mathrm{\ell }(x_k)$. The refinement condition is a condition on supports, not an ordering of incomparable quotients of equal length; only the lengths are ordered. For $i=\mathrm{\ell }(x)$ the graded multiplicity is $(M:{\mathrm{\Delta }}_x(d)):=\text{multiplicity of }{\mathrm{\Delta }}_x(d)\text{ in }{\mathrm{\Gamma }}_{\ge i}M\mathrm{/}{\mathrm{\Gamma }}_{\ge i+1}M,$ and, for a $\mathrm{\nabla }$-flag, $(M:{\mathrm{\nabla }}_x(d))$ is the multiplicity of ${\mathrm{\nabla }}_x(d)$ in ${\mathrm{\Gamma }}_{\le i}M\mathrm{/}{\mathrm{\Gamma }}_{\le i-1}M$; a priori both symbols refer to a chosen compatible enumeration (the next items on this page prove that they do not depend on it). With $T_x$ the Hecke basis of The standard basis of the type-A Hecke algebra and its multiplication rule and ${\tilde{T}}_x:=v^{\mathrm{\ell }(x)}{T}_x$, the character sums attached to a flag are $h_{\mathrm{\Delta }}(M):={\sum }_{x,d}(M:{\mathrm{\Delta }}_x(d))v^d{\tilde{T}}_x,h_{\mathrm{\nabla }}(M):={\sum }_{x,d}(M:{\mathrm{\nabla }}_x(d))v^{-d}{\tilde{T}}_x,$ and they are used only through the items that prove these sums intrinsic.

Small $n$. For $n\le 1$ there is a single element $e$ and every object is a finite direct sum of shifts ${⨁}_{j}R(d_j)$. Its one-graph flags have these summands as quotients, so $h_{\mathrm{\Delta }}={\sum }_j{v}^{d_j}{\tilde{T}}_e$ and $h_{\mathrm{\nabla }}={\sum }_j{v}^{-d_j}{\tilde{T}}_e$; in particular both characters send the unit $R$ to $1$.

(a) Maps between standard bimodules. For $v,w\in S_n$ and $a,b\in \mathbb{Z}$, ${\mathrm{Hom}}_{R\text{-}R}(R_v(a),R_w(b))\cong \{\begin{array}{ll}R(b-a),& v=w,\\ 0,& v\mathrm{\ne }w,\end{array}$ where $R(c)$ denotes the free graded $R$-module on one generator of degree $-c$. Indeed a bimodule map is determined by $\phi (1)=h\in R$, and $f\cdot 1\cdot g\mapsto fhw(g)$ must equal $v(g)fh$; since $R$ is a domain this forces $v(g)h=w(g)h$ for all $g$, hence $h=0$ or $v=w$. The generator $1$ of $R_v(a)$ has degree $-a$ and maps to $h$, which has degree $\mathrm{deg}h-b$ as an element of $R_w(b)$, so the homogeneous map ${\phi }_h$ has degree $\mathrm{deg}h+a-b$ and $\mathrm{Hom}(R_v(a),R_w(b))\cong R(b-a)$ in the case $v=w$.

(b) Composition. The $R$-balanced assignment $R_w{\otimes }_R{R}_v⟶R_{wv},a\otimes b⟼aw(b),$ where $w(b)(\lambda )=b(w^{-1}\lambda )$ and the product is taken in $R$, is a well-defined degree-zero isomorphism of graded $(R,R)$-bimodules. It is well defined because for $f\in R$ the relation $(a\cdot f)\otimes b=a\otimes (fb)$ reads $w(f)a\otimes b=a\otimes fb$ and maps to $w(f)aw(b)=aw(f)w(b)$ on both sides; it is left $R$-linear, and it respects the right actions because $w(bh)=w(b)w(h)$ for $h\in R$. It sends the $1$-tensor $1\otimes 1$ of degree $0$ to $1$ and is therefore surjective, and both sides are free of rank one as left $R$-modules, so it is an isomorphism. In particular $R_w{\otimes }_R{R}_v\cong R_{wv}$ (and $R_w{\otimes }_R{R}_v\cong R_w$ only when $v=e$), while tensoring with $w=e$ shows that the assignment is compatible with the unit. The single assignment $1\otimes 1\mapsto 1$ does extend uniquely to a bimodule map, and the map it extends to is exactly the twisted one above; what fails is the untwisted formula $a\otimes b\mapsto ab$, which is not balanced when $w\mathrm{\ne }e$: the equality $(1\cdot f)\otimes 1=1\otimes f$ in $R_w{\otimes }_R{R}_v$ would map to $w(f)$ on the left and $f$ on the right, and these differ for a polynomial moved by $w$.

(c) The rank-one calculus. Let $s=s_i$, $R^s=R^{s_i}$, $\alpha ={\alpha }_i$, $\delta :=\alpha \mathrm{/}2$, and $B_s=R{\otimes }_{R^s}R(1)$ with the images $u:=1\otimes 1$ of degree $-1$ and $w_0:=1\otimes \delta$ of degree $1$ (The Soergel bimodule $B_i$ of a simple reflection); these two elements form a graded left $R$-basis of $B_s$, and every $g\in R$ has a unique decomposition $g=g^{+}+\delta g^{-}$ with $g^{\pm }\in R^s$, given by $g^{+}=\frac{1}{2}(g+s(g))$. The right action on the basis is $u\cdot g=g^{+}u+g^{-}{w}_0,w_0\cdot g={\delta }^2{g}^{-}u+g^{+}{w}_0,$ as follows from $1\otimes g=1\otimes (g^{+}+\delta g^{-})=g^{+}u+g^{-}{w}_0$ and $\delta g=\delta g^{+}+{\delta }^2{g}^{-}$ with ${\delta }^2={\alpha }^2\mathrm{/}4\in R^s$. Hence $(\delta u+w_0)\cdot g=g(\delta u+w_0),(\delta u-w_0)\cdot g=s(g)(\delta u-w_0),$ so $R(\delta u+w_0)\cong R(-1)$ and $R(\delta u-w_0)\cong R_s(-1)$ are sub-bimodules of $B_s$, generated in degree $1$. Moreover for $f,h\in R$ one has $(fu+h{w}_0)\mapsto f+h\delta$ under $f\otimes g\mapsto fg$ and $(fu+h{w}_0)\mapsto f-h\delta$ under $f\otimes g\mapsto fs(g)$, because $1\otimes g=g^{+}u+g^{-}{w}_0$ and $s(g)=g^{+}-\delta g^{-}$; both maps are surjective, their kernels are the two displayed sub-bimodules, and all four maps of $0\to R(-1)\stackrel{1\mapsto \delta u+w_0}{\to }{B}_s\stackrel{f\otimes g\mapsto fs(g)}{\to }{R}_s(1)\to 0,$ $0\to R_s(-1)\stackrel{1\mapsto \delta u-w_0}{\to }{B}_s\stackrel{f\otimes g\mapsto fg}{\to }R(1)\to 0$ are degree zero: sources and targets are generated in degree whose shifts match the degree-$1$ elements $\delta u\pm w_0\in B_s$. Finally the two quotients are the image of $u$, a free rank-one $R$-module with right action $u\cdot g\equiv gu$ modulo $\delta u-w_0$ and $u\cdot g\equiv s(g)u$ modulo $\delta u+w_0$ respectively, that is $R(1)$ and $R_s(1)$.

(d) Structural proofs. The definition fixes graph bimodules, canonical support flags and character conventions. The following items supply their structural results locally: Bott–Samelson bimodules carry delta and nabla support filtrations proves generator closure and both flag recursions; Special Bott–Samelson Hom formula before reflection localization proves the two flagged Bott–Samelson Hom cases; Type-A top support layers are controlled by reflection localization proves the four-layer freeness, reflection-localized image equalities and right splitting; and The type-A Soergel Hom formula extends the flagged Hom calculation to arbitrary Soergel summands. These links locate proofs rather than adding imported theorems to this definition's assumptions.