# Tuesday July 14th :::{.remark} \hfill - The KL basis can be complicated -- you can compute them recursively, but may need to subtract off "constant coefficients" to obtain self-dual elements. - There are no closed formulas in general. - For any $P\in ! + q \ZZ^{\geq 0}[q]$ there exists an $m\in \ZZ$ such that $v^m P(v^{-2})$ occurs as a KL-polynomial. - **The KL Positivity Conjecture**: $h_{x, w} \in \ZZ^{\geq 0} [v]$, and in fact these non-negative coefficients can be realized as the graded dimension of the local intersection cohomology of Schubert varieties. ::: ## Why care about KL Positivity? Set \[ \lieg = \liesl(n, \CC) = \theset{A\in \CC^{n\times n} \suchthat \tr(A) = 0}\\ \text{with } [AB] = AB - BA .\] The case of finite-dimensional representations is well understood, since everything decomposes into simple modules. In the infinite case, we consider the principal block of category $\OO$, denoted $\OO_0(\lieg)$. There is a correspondence \begin{tikzcd} \correspond{\text{Simple modules in } \OO_0} & \ar[l] S_n \ar[r] & \correspond{\text{Verma modules in }\OO_0} \\ \parbox{3cm}{\centering Weight modules $L(w)$ } & \ar[l] w \ar[r] & \Delta(w) \end{tikzcd} The **KL-multiplicity conjecture** states \[ [\Delta(w) : L(x)] = h_{x, w}(1) ,\] where the LHS counts how often $L(x)$ occurs as a subquotient in a composition series for $\Delta(w)$. Determining the characters $\ch L(w)$ can be done using translation functors, and corresponds to determining the characters of all simple highest weight modules (not necessarily finite-dimensional), and is a vast generalization of Weyl's character formula. This was the birth of geometric representation theory, and the proof involved $D\dash$modules, perverse sheaves, and Deligne's proof of the Weil conjectures. **Goal**: categorify $\mch_n(S_n)$, the associative unital $\ZZ[v, v\inv]\dash$algebra given by $\gens{\theset{H_i \suchthat i\leq n-1}}$ subject to \[ H_{s_{i+1}} H_{s_{i}} H_{s_{i+1}} &= H_{s_{i+1}} H_{s_{i}} H_{s_{i+1}} \\ H_{s_{i}} H_{s_{j}} &= H_{s_{j}} H_{s_{i}} && \abs{i-j} \geq 2 \\ H_{s_{i}}^2 &= (v\inv - v) H_{s_i} + 1 .\] Or equivalently, setting $C_{s_i} = H_{s_i} + v$, \[ C_{s_{i+1}} C_{s_{i}} C_{s_{i+1}} + C_{s_i} &= C_{s_{i+1}} C_{s_{i}} C_{s_{i+1}} + C_{s_{i+1}} \\ C_{s_{i}} C_{s_{j}} &= C_{s_{j}} C_{s_{i}} &&\abs{i-j} \geq 2 \\ C_{s_{i}}^2 &= (v\inv - v) H_{s_i} + 1 .\] :::{.remark} The presentation above can be "lifted" to a categorical level. So we want to find a category $\mca$ which is additive, monoidal, graded, and abelian such that - $K_0^\oplus(\mca) \cong H(S_n)$ is an isomorphism of $\ZZ[v, v\inv]\dash$algebras. - There are objects $B_{s_i}$ such that \[ 1. && B_{s_i} \tensor B_{s_i} &\cong B_{s_i}(1) \oplus B_{s_i}(-1) \\ 2. && B_{s_j} \tensor B_{s_i} &\cong B_{s_i} \tensor B_{s_j} && \text{ for } \abs{i-j} \geq 2 \\ 3. && B_{s_i} \tensor B_{s_{i+1}} \tensor B_{s_i} \oplus B_{s_{i+1}} &\cong B_{s_{i+1}} \tensor B_{s_{i}} \tensor B_{s_{i+1}} \oplus B_{s_{i}} .\] ::: Question 1: What are the objects $B_{s_i}$? Set $R = \CC[x_1, \cdots, x_n]$ with $\abs{x_i} = 2$, yielding a graded $\CC\dash$algebra where $s_i$ permutes $x_i, x_{i+1}$. We can look at the invariant ring, \[ R^{s_i} \definedas \theset{f\in R \suchthat s_i f = f} \] and note that $R$ and any of its shifts are modules over this ring. So set, \[ B_{s_i} \definedas R \tensor_{R^{s_i}} R(-1) \] which is a graded $R\dash$bimodule. :::{.proposition title="?"} We have an isomorphism of $R\dash$bimodules satisfying the desired relations. ::: :::{.claim} $R\cong R^{s_i}(2) \oplus R^{s_i}$ as graded $R^{s_i}\dash$bimodules. ::: :::{.proof} It suffices to show that every $f\in R$ can be written uniquely as \[ f = g(x_i - x_{i+1}) + h \text{ with } g, h \in R^{s_i} .\] \ **Uniqueness**: If $f =g'(x_i - x_{i+1}) + h'$ then \[ g'(2x_i - 2x_{i+1}) &= f - s_i f \\ &= g(x_i - x_{i+1}) - g(x_{i+1} - x_i) \\ &= g(x_i - x_{i+1} - x_{i+1} + x_i) \\ &= g(2x_i - 2x_{i+1}) .\] Since this is an integral domain, $g=g'$, and \[ h = f - g (x_i - x_{i+1}) = f - g'(x_i - x_{i+1}) = h' .\] **Existence**: $x_k \in R^{s_i}$ is $s_i$ invariant if $k\neq i, i+1$, and \[ x_i = {1\over 2}\qty{ x_i - x_{i+1}} + {1\over 2}\qty{x_i + x_{i+1}} \in R^{s_i} \qty{\cdots} + R^{s_i} \\ x_{i+1} = -{1\over 2}\qty{ x_i - x_{i+1}} + {1\over 2}\qty{x_i + x_{i+1}} \in R^{s_i} \qty{\cdots} + R^{s_i} \\ .\] So every $f\in R$ can be expressed as a polynomial in $x_i - x_{i+1}$ with coefficients in $R^{s_i}$. Since $\qty{x_i - x_{i+1}}^2 \in R^{s_i}$, any polynomial in $x_i - x_{i+1}$ with coefficients in $R^{s_i}$ can be written as $g(x_i - x_{i+1}) + h$ with $g, h \in R^{s_i}$. This proves the claim. We can now check \[ B_{s_i} \tensor_R B_{s_i} &= \qty{R \tensor_{R^{s_i}} R} \tensor_R \qty{R \tensor_{R^{s_i}} R}(-2) \\ &\cong R \tensor_{R^{s_i}} R \tensor_{R^{s_i}} R(-2) \\ &\cong R \tensor_{R^{s_i}} \qty{ R^{s_i}(2) \oplus R^{s_i} } \tensor_{R^{s_i}} R(-2) \\ &\cong R \tensor_{R^{s_i}} \qty{ R^{s_i}(2) } \tensor_{R^{s_i}} R(-2) \oplus R \tensor_{R^{s_i}} \qty{ R^{s_i} } \tensor_{R^{s_i}} R(-2) \\ &\cong R \tensor_{R^{s_i}} R \oplus R\tensor_{R^{s_i}} R(-2) \\ &\cong B_{s_i}(1) \oplus B_{s_i}(-1) .\] ::: ## Soergel's Dream Come up with a purely algebraic proof of the KL conjecture (without using machinery from geometric representation theory). Consider the center of category $O$, $Z(\OO_0)$, the endomorphism ring of the identity functor. It can be shown that this is isomorphic to the coinvariant ring $\CC[x_1, \cdots, x_{n+1}] / \CC[x_1, \cdots, x_{n+1}]^{S^n}$. Thus invariant theory is "hidden" in the category $\OO_0(\lieg)$. > Proved originally, but used some decomposition theory. > Recent proof from Elias Williams? Using hodge structures. **Question**: What is the category $\mca$? :::{.definition title="Bott-Samelson Bimodules"} Take $R\dash\grdim$, the category of graded $R\dash$bimodules, which is finitely generated as both left/right $R\dash$modules. This is additive, monoidal, and graded, but this category is too big. So we carve out a smaller subcategory. For $w\in S_n$, write the formal word $\bar w = s_{i_1} \cdots s_{i_r}$ a reduced expression for $w$. Note that this depends on which reduced expression is used. Now define \[ BS(\bar w) &\definedas \bigotimes^{j\leq r}{}_{R^{s_{i_j}}} \,\,B_{s_{i_j}} && BS(\emptyset) = R \\ &\cong R \tensor_{R^{s_{i_1}}} R \tensor \cdots \tensor_{R^{s_{i_r}}} R \] where the isomorphism is canonical. This is the **Bott-Samelson bimodule**. ::: :::{.definition title="?"} \hfill - A Soergel bimodule is a direct summand of a finite sum of grading shifts of Bott-Samelson bimodules. - The category $\mathbf{SBim}$ of Soergel bimodules is the strictly full subcategory of $R\dash\grdim$ consisting of Soergel bimodules. - Equivalently, the smallest full subcategory of $R\dash\grdim$ consisting of $R, B_{s_i}$ and closed under $\tensor_R, \oplus, (i)$ and taking direct summands. ::: :::{.remark} The category $\mathbf{SBim}$ is additive, monoidal, and graded, but **not** abelian. ::: :::{.definition title="Indecomposable"} Recall that a module $X$ is **indecomposable** $\iff$ $X \cong A\oplus B$ implies $A\cong 0$ or $B\cong 0$. ::: :::{.definition title="Krull-Schmidt"} A category $\mathcal{C}$ is **Krull-Schmidt** $\iff$ every object decomposes uniquely into a *finite* direct sum of indecomposable objects. ::: :::{.theorem title="?"} \hfill 1. $\mathbf{SBim}$ is *Krull-Schmidt*. > Note that such uniqueness here means that if $\bigoplus_{i=1}^r X_i \cong \bigoplus_{i=1}^s Y_i$ then $r=s$ and there is some permutation $\pi \in S_r$ such that $X_{\pi(i)} \cong Y_i$. 2. There is a bijection \[ S_n &\tofrom \correspond{\text{Indecomposable Soergel bimodules} }/ \text{\tiny Isomorphism and shifts} \\ w &\mapsto B_w .\] Moreover $\theset{ S[B_w] \suchthat w\in S_n }$ form a basis of $K_0^\oplus(\mathbf{SBim})$ as $\ZZ[v, v\inv]\dash$modules. 3. There is a $\ZZ[v, v\inv]\dash$algebra morphism \[ c: \mch(S_n) &\mapsvia{\cong} K_0^\oplus(\mathbf{SBim}) \\ C_{s_i} &\mapsto [B_{s_i}] .\] ::: :::{.remark} This implies both the KL positivity and multiplicity conjectures. The KL basis corresponds to the basis given by the indecomposable Soergel bimodules. :::