\documentclass[revised]{FPSAC2017} \articlenumber{63} \addbibresource{63_Kvinge_Licata_Mitchell.bib} %------------------------------- Various packages ----------------------------------------------------------------------------- \usepackage[dvipsnames]{xcolor} \usepackage{listings, hyperref, multicol, shuffle, xcolor, enumerate, dsfont, tikz, dsfont, tabu, graphicx,mathtools,array} \usepackage{mathabx} \usepackage{tikz} \usetikzlibrary{decorations.pathreplacing} \usepackage{tikz} %------------------------------- Theorem environments ----------------------------------------------------------------------------- \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{corollary}[theorem]{Corollary} %------------------------------- Tikz Related 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End_{\MC{H'}}(\UnitModule) %%% \newcommand{\partsofpartition}[2]{m_{#1}(#2)} %%% Parts equal to x of partition y, m_x(y) for x,y variables %%% \newcommand{\fallingfactorial}[2]{( #1 \downharpoonright #2)} %%% Falling factorial, (x \downharpoon y) for x,y variables %%% \newcommand{\content}[1]{\text{cont}(#1)} %%% Content of cell in YD, cont(x) for x variable %%% \newcommand{\transition}[1]{\wh{\omega}_{#1}} %%% Transition measure, \widehat{omega_x} for x variable %%% \newcommand{\cotransition}[1]{\widecheck{\omega}_{#1}} %%% Co-transition measure, \widecheck{\omega_x} for x variable %%% \newcommand{\transitionprob}[2]{\wh{q}_{#1}(#2)} %%% Transition probability, p_x(y) for x,y variables %%% \newcommand{\cotransitionprob}[2]{\widecheck{q}_{#1}(#2)} %%% Co-transition probability, q_x(y) for x,y variables %%% \newcommand{\moment}[1]{\wh{m}_{#1}} %%% Transition measure moments, \widehat{\sigma_x} for x variables %%% \newcommand{\comoment}[1]{\widecheck{m}_{#1}} %%% Co-transition measure 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h^*_x for x variable %%% \newcommand{\contentsalpha}[1]{A_{#1}} %%% Algebra of contents for a diagram %%% \newcommand{\symalgcontent}[1]{S(A_{#1})} %%% Symmetric algebra of content of partition, S(A_x) for x variable %%% \newcommand{\contentmap}{\gamma} %%% Map from symmetric functions to shifted symmetric functions by content, \gamma %%% \newcommand{\Heisencat}{\MC{H}'} %%% Heisenberg category, H' %%% \newcommand{\SymtoHplus}[1]{\MC{T}_{#1}} %%% Map from symmetric group to End_H'(Q^+), \MC{T}_x for x variable %%% \newcommand{\EndUpStrands}[1]{\End_{\MC{H}'}(Q_{+^{#1}})} %%% Endomorphism algebra of x upward pointed strands, \End_{\MC{H}'}(Q_{+^{x}}) for x variable %%% \newcommand{\cgenfunction}{c(z)} %%% Generating function for c_k's, c(z) %%% \newcommand{\ctildegenfunction}{\tilde{c}(z)} %%% Generating function for \tilde{c}_k's, c(z) %%% \newcommand{\YoungIdempotent}[1]{E_{#1}} %%% Young idempotent, e_x, for x variable %%% \newcommand{\JM}[1]{J_{#1}} %%% Jucys-Murphy element, J_x, for x variable %%% \newcommand{\ckimage}[2]{c_{#1,#2}} %%% Image of clockwise bubble under F, c_{x,y} for x,y variables %%% \newcommand{\cktildeimage}[2]{\tilde{c}_{#1,#2}} %%% Image of counterclockwise bubble under F, \tilde{c}_{x,y} for x,y variables %%% \newcommand{\pr}[1]{\text{pr}_{#1}} %%% Projection operator from S_n to S_{n-1}, pr_x for x variable %%% \newcommand{\disturbdeg}{\deg} %%% Disturbance filtration "degree", \gamma_{dist} %%% \newcommand{\disturb}[1]{F_{#1}} %%% Disturbance filtered part, disturb_x for x variable %%% \newcommand{\Youngidempotent}[1]{E_{#1}} %%% Young idempotent, E_x for x variable %%% \newcommand{\expectation}{\wh{\MB{E}}} %%% Expectation value for transition measure, \wh{\MB{E}} %%% \newcommand{\coexpectation}{\widecheck{\MB{E}}} %%% Expectation value for co-transition measure, \widecheck{\MB{E}}} %%% \newcommand{\integralHeisen}{H_{\MB{Z}}} %%% Integral form of Heisenberg, H_{\MB{Z}} %%% \newcommand{\categorificationmap}{\phi} %%% Categorification map, \phi %%% \newcommand{\involution}{\xi} %%% Involution on \Heisencat, \xi %%% \newcommand{\crossingplusdots}{c} %%% Number of crossings and dots of diagram, c %%% \newcommand{\shiftinvolution}{I} %%% Involution of shifted symmetric functions, I %%% \newcommand{\interlacingx}[1]{a_{#1}} %%% Interlacing coordinate for minimum, a_x for x variable %%% \newcommand{\interlacingy}[1]{b_{#1}} %%% Interlacing coordinate for maximum, b_x for x variable %%% \newcommand{\Walgebra}{W_{1 + \infty}} %%% W_{1 + \infty) %%% \newcommand{\FH}{\MC{K}} %%% FarahatHigmanRing, \MC{K} %%% \newcommand{\distsym}{\gamma_{\text{dist}}'} %%% Disturbance degree function for symmetric groups, \gamma_{dist}' %%% \newcommand{\distpart}{\gamma_{\text{dist}}''} %%% Disturbance degree function partitions, \gamma_{dist}'' %%% \newcommand{\contentnumb}[2]{\text{cont}_{#1}(#2)} %%% Number contents of cells %%% \newcommand{\arrow}{ \begin{tikzpicture} \draw[thick] (3,0)--(3.08,.08); \draw[thick] (3,0)--(3.08,-.08); \end{tikzpicture} } %%% Arrow picture %%% %------------------------------- Extra New Commands ------------------------------------------------------------------------------------------ \newcommand{\omitt}[1]{} \newcommand{\tonytodo}{\todo[inline,color=pink!20]} \newcommand{\henrytodo}{\todo[inline,color=lightblue]} %------------------------------- Front Page Matters ------------------------------------------------------------------------------------------ \title[Khovanov's Heisenberg category and shifted symmetric functions]{Khovanov's Heisenberg category, moments in free probability, and shifted symmetric functions} \author[H. Kvinge, A. Licata, S. Mitchell]{Henry Kvinge\addressmark{1}, Anthony M. Licata\thanks{AML was supported by a Discovery Project grant from the Australian Research Council.}\addressmark{2} \and Stuart Mitchell\addressmark{2}} \address{\addressmark{1}Department of Mathematics, UC Davis, Davis, CA, USA \\ \addressmark{2}Mathematical Sciences Institute, Australian National University, Canberra, Australia} \received{\today} \abstract{We establish an isomorphism between the center $\text{End}_{\mathcal{H}'}(\mathbf{1})$ of Khovanov's Heisenberg category $\mathcal{H}'$ and the algebra $\Lambda^*$ of shifted symmetric functions defined by Okounkov-Olshanski. We give a graphical description of the shifted power and Schur bases of $\Lambda^*$ as elements of $\text{End}_{\mathcal{H}'}(\mathbf{1})$, and describe the curl generators of $\text{End}_{\mathcal{H}'}(\mathbf{1})$ in the language of shifted symmetric functions. This latter description makes use of the transition and co-transition measures of Kerov and the noncommutative probability spaces of Biane.} \keywords{shifted symmetric functions, Heisenberg algebra categorification, asymptotic representation theory of symmetric groups, noncommutative probability theory} \begin{document} \maketitle \section{Introduction} In \cite{Kho14}, Khovanov introduces a graphical calculus of oriented planar diagrams and uses it to define a linear monoidal category $\Heisencat$, which he proposes as a categorification of the Heisenberg algebra. We denote by $\Hcenter$ the endomorphism algebra of the monoidal unit in $\Heisencat$. The commutative algebra $\Hcenter$ is, by definition, the algebra of closed oriented planar diagrams modulo the relations of the Khovanov graphical calculus. In his study of morphism spaces of $\Heisencat$, Khovanov introduces two sets of generators for $\Hcenter$: the clockwise curls $\{c_k\}_{k \geq 0}$ and the counterclockwise curls $\{\ctilde{k}\}_{k\geq 2}$. He then establishes algebra isomorphisms $$ \Hcenter \cong \MB{C}[c_0,c_1,c_2, \dots] \cong \MB{C}[\ctilde{2},\ctilde{3},\ctilde{4}, \dots], $$ and describes a recursion for expressing the clockwise and counterclockwise curls in terms of each other. He then relates $\Heisencat$ to representation theory by defining a sequence of monoidal functors $\htobimod{k}$ from $\Heisencat$ to bimodule categories for symmetric groups. A consequence of the existence of these functors is the existence of surjective algebra homomorphisms, $$ f_n^{\Heisencat} : \Hcenter \longrightarrow Z(\MB{C}[\Sy{n}]), $$ from $\Hcenter$ to the center of the group algebra of each symmetric group. Based in part on this, Khovanov suggests that there should be a close connection between $\Hcenter$ and the asymptotic representation theory of symmetric groups. Furthermore, one might hope that $\Hcenter$ in fact gives a diagrammatic description of some algebra of pre-existing combinatorial interest. The main goal of the current paper is to make precise the connection between $\Hcenter$ and both the asymptotic representation theory of symmetric groups and algebraic combinatorics. We do this by establishing an isomorphism between $$ \primaryiso : \Hcenter \longrightarrow \ShiftSym{}, $$ where $\ShiftSym{}$ is the {\emph{shifted symmetric functions}} of Okounkov-Olshanski \cite{OO97}. (See \cref{thm-main-1}.) The algebra of shifted symmetric functions $\ShiftSym{}$ is a deformation of the algebra of symmetric functions. As is the case for $\Hcenter$, there are surjective algebra homomorphisms $$ f_n^{\ShiftSym{}}: \ShiftSym{} \longrightarrow Z(\MB{C}[\Sy{n}]), $$ to the center of the group algebra of each symmetric group. The isomorphism $\primaryiso : \Hcenter \longrightarrow \ShiftSym{}$ is canonical, in that it intertwines the homomorphisms $ f_n^{\Heisencat} $ and $f_n^{\ShiftSym{}}$. The isomorphism $\primaryiso:\Hcenter \longrightarrow \ShiftSym{}$ allows us to give a graphical description of several important bases of $\ShiftSym{}$. For example, the shifted power sum denoted $p_\lambda^\#$ in \cite{OO97} appears in $\Hcenter$ as the closure of a permutation of cycle type $\lambda$. The shifted Schur function $s_\lambda^*$ appears as the closure of a Young symmetrizer of type $\lambda$. (See \cref{cor-Schur}). In the other direction, it is also reasonable to ask for a description of the image of Khovanov's curl generators $c_k$ and $\ctilde{k}$ as elements of $\ShiftSym{}$. It turns out that the right language for such a description is that of noncommutative probability theory. In \cite{Ker93}, Kerov introduces, for each partition $\lambda$, a pair of finitely supported probability measures on $\MB{R}$; these probability measures are known as the \emph{ transition} and {\emph{co-transition} measures, or sometimes as growth and decay. In work of Biane \cite{B98}, these probability measures appear as the compactly-supported measures associated to self-adjoint operators on a noncommutative probability space, and as a result they are basic objects of interest at the intersection of representation theory and noncommutative probability theory. In particular, the \emph{moments} and \emph{Boolean cumulants} of the transition and co-transition measures may be regarded as elements of $\ShiftSym{}$. In \cref{thm-moment-images}, we show that the isomorphism $\primaryiso$ takes Khovanov's curl generators $c_k$ and $\ctilde{k}$ to scalar multiples of the $k$th moments of Kerov's transition and co-transition measures. In fact, the close relationship between the transition and co-transition measures themselves yields two independent descriptions of the image of the curl generator $c_k$: it is equal to a scalar multiple of both the $k$th moment of the co-transition measure and the $(k+2)$th Boolean cumulant of the transition measure. The observation that the Boolean cumulants of the transition measure are equal to the moments of the co-transition measure seems to be new, and is closely connected to the adjointness of induction and restriction functors between representation categories of symmetric groups. A dictionary between several of the bases of $\Hcenter$ and $\ShiftSym{}$ is given in \cref{i-class-table}. The existence of a relationship between $\Heisencat$ and free probability -- and indeed, much of this paper -- was anticipated by Khovanov in \cite{Kho14}. The relationship between generators of $\Hcenter$ and the noncommutative probability spaces of \cite{B98} may be seen as a further manifestation of the ``planar structure" of free probability; the many connections between noncommutative probability and other mathematical subjects with planar structure are emphasized in the work of Guionnet, Jones and Shlyakhtenko \cite{GJS10}. This text is an extended abstract of the preprint \cite{KLM16}, where complete proofs and additional background can be found. \vspace{-3mm} \section{The symmetric group and its normalized character theory} \label{sect-sym-group} We begin by establishing notation related to partitions and Young diagrams. Let $\partitionsn{n}$ be the set of partitions of $n$ and $\partitionsn{} := \bigcup_{n \geq 0} \partitionsn{n}.$ We freely identify $\mu \in \partitions$ with its corresponding Young diagram, which we draw using Russian notation (see \cref{example-interlacing-sequences}). If $\ydcell$ is a cell in the $i$th row and $j$th column of $\mu$ then the {\emph{content}} of $\ydcell$ is defined as $\content{\ydcell} := j-i$. We say that a cell $\ydcell \notin \mu$ is {\emph{$i$-addable}} with respect to $\mu$ if it has content $i$ and adding it to $\mu$ gives a Young diagram. We say that a cell $\ydcell \in \mu$ is {\emph{$i$-removable}} with respect to $\mu$ if it has content $i$ and removing it from $\mu$ gives a Young diagram. We call two sequences $\interlacingx{1}, \dots, \interlacingx{d}$ and $\interlacingy{1}, \dots, \interlacingy{d-1}$ {\emph{interlacing}} when \begin{equation*} \interlacingx{1} < \interlacingy{1} < \interlacingx{2} < \dots < \interlacingx{d-1} < \interlacingy{d-1} < \interlacingx{d}. \end{equation*} The \emph{center} of this pair of sequences is defined as the quantity $(\interlacingx{1} + \dots + \interlacingx{d}) - (\interlacingy{1} + \dots + \interlacingy{d-1})$. There is a bijection between Young diagrams and pairs of integer-valued interlacing sequences $\interlacingx{1}, \dots, \interlacingx{d}$ and $\interlacingy{1}, \dots, \interlacingy{d-1}$ with center 0. Given $\mu$ the corresponding sequence $\interlacingx{1}, \dots, \interlacingx{d}$ is the ordered list of all $\interlacingx{j}$ such that there exists an $\interlacingx{j}$-addable cell with respect to $\mu$, while $\interlacingy{1}, \dots, \interlacingy{d-1}$ is the ordered list of all $\interlacingy{j}$ such that there exists a $\interlacingy{j}$-removable cell with respect to $\mu$. It is clear that $\interlacingx{1}, \dots, \interlacingx{d}$ and $\interlacingy{1}, \dots, \interlacingy{d-1}$ are interlacing. We denote by $\mu^{(j)}$ the Young diagram that we obtain by adding a cell of content $\interlacingx{j}$, so that $\content{\mu^{(j)} / \mu} = \interlacingx{j}$. Similarly, we denote by $\mu_{(j)}$ the Young diagram that we obtain by removing a cell of content $\interlacingy{j}$ from $\mu$, so that $\content{\mu / \mu_{(j)}} = \interlacingy{j}$. \begin{example} \label{example-interlacing-sequences} Let $\mu = (4,2,1)$. Then $\mu$ yields the interlacing sequences \begin{equation*} {\color{red}{-3}} < {\color{red}{-1}} < {\color{red}{1}} < {\color{red}{4}} \quad\quad \text{and} \quad\quad {\color{blue}{-2}} < {\color{blue}{0}} < {\color{blue}{3}}. \end{equation*} \begin{center} \begin{tikzpicture}[scale = .9] \vspace{3mm} %labelled diagonals (above) \node at (2.8,2.3) {\color{red}{\scriptsize{$\interlacingx{4}$}}}; \node at (2.1,2.3) {\color{blue}{\scriptsize{$\interlacingy{3}$}}}; \node at (.7,2.3) {\color{red}{\scriptsize{$\interlacingx{3}$}}}; \node at (0,2.3) {\color{blue}{\scriptsize{$\interlacingy{2}$}}}; \node at (-.7,2.3) {\color{red}{\scriptsize{$\interlacingx{2}$}}}; \node at (-1.4,2.3) {\color{blue}{\scriptsize{$\interlacingy{1}$}}}; \node at (-2.1,2.3) {\color{red}{\scriptsize{$\interlacingx{1}$}}}; \draw (-4,-2.00) -- (4,-2.00); %labels on diagonals (below) \node at (2.72,-2.2) {\scriptsize{$4$}}; \node at (2.05,-2.2) {\scriptsize{$3$}}; \node at (1.37,-2.2) {\scriptsize{$2$}}; \node at (.7,-2.2) {\scriptsize{$1$}}; \node at (0,-2.2) {\scriptsize{$0$}}; \node at (-.78,-2.2) {\scriptsize{$-1$}}; \node at (-1.48,-2.2) {\scriptsize{$-2$}}; \node at (-2.15,-2.2) {\scriptsize{$-3$}}; \node at (0,0) {\begin{tikzpicture}[rotate = 45, scale = 1] %first row diagram \draw (0,0) rectangle (.8,.8); \draw (.8,0) rectangle (1.6,.8); \draw (1.6,0) rectangle (2.4,.8); \draw (2.4,0) rectangle (3.2,.8); %second row diagram \draw (0,.8) rectangle (.8,1.6); \draw (.8,.8) rectangle (1.6,1.6); %third row diagram \draw (0,1.6) rectangle (.8,2.4); %coordinate axises \draw[->] (0,0) -- (0,5); \draw[->] (0,0) -- (5,0); %diagonals above y = x \draw[dotted] (0,0) -- (2.4,2.4); \draw[dotted] (-.4,.4) -- (2,2.8); \draw[dotted] (-.8,.8) -- (1.6,3.2); \draw[dotted] (-1.2,1.2) -- (1.2,3.6); %diagonals below y = x \draw[dotted] (.4,-.4) -- (2.8,2); \draw[dotted] (.8,-.8) -- (3.2,1.6); \draw[dotted] (1.2,-1.2) -- (3.6,1.2); \draw[dotted] (1.6,-1.6) -- (4,.8); \end{tikzpicture}}; \end{tikzpicture} \end{center} \end{example} Let $\Sy{n}$ be the symmetric group with Coxeter generators $s_1, \dots, s_{n-1}$. If $g \in \Sy{n}$ has cycle type $\lambda \in \partitionsn{n}$, then we write $\shape{g} := \lambda$. For $k \leq n$, there is an embedding ${\symembedding{k}{n}:\MB{C}[\Sy{k}] \hookrightarrow \MB{C}[\Sy{n}]}$ called the {\emph{standard embedding}} which sends $\Sy{k}$ to the subgroup generated by $s_1, \dots, s_{k-1}$. Let $\simplerep{\lambda}$ be the simple $\MB{C}[\Sy{n}]$-module (i.e. the irreducible $\Sy{n}$ representation) associated to $\lambda \in \partitionsn{n}$ and $\charrep{\lambda}: \MB{C}[\Sy{n}] \rightarrow \MB{C}$ its character. Abusing notation, we write $\charrep{\lambda}(\mu)$ for $\charrep{\lambda}(g)$ when $\shape{g} = \mu$. The {\emph{normalized character}} $\norcharrep{\lambda}: \bigoplus_{k \leq n} \MB{C}[\Sy{k}] \rightarrow \MB{C}$ associated to $\lambda$ is defined so that for $x \in \MB{C}[\Sy{k}]$, \begin{equation} \label{eqn-char-map} \norcharrep{\lambda}(x) := \frac{\charrep{\lambda}(\symembedding{k}{n}(x))}{\dim \simplerep{\lambda}} = \frac{\charrep{\lambda}(\symembedding{k}{n}(x))}{\charrep{\lambda}(1)}. \end{equation} \begin{definition} \label{def-identity-of-a} For $\mu = (\mu_1, \dots, \mu_t) \in \partitionsn{k}$ with $k \leq n$, set \begin{equation} \label{eqn-alt-a} \classsum{\mu}{n} = \sum (i_1,\dots, i_{\mu_1}) \dots (i_{k-\mu_t +1}, \dots, i_k) \end{equation} where this sum is taken over all distinct $k$-tuples $(i_1, \dots, i_k)$ of elements from $\{1, 2, \dots, n\}$. We call $\classsum{\mu}{n}$ the {\emph{normalized conjugacy class sum}} associated to $\mu$ in $\MB{C}[\Sy{n}]$. \end{definition} The elements $\classsum{\mu}{n}$ belong to $Z(\MB{C}[\Sy{n}])$ and for $\lambda \in \partitionsn{n}$ \begin{equation} \label{prop-normalized-char-class-sum} \norcharrep{\lambda}(\classsum{\mu}{n}) = (n \downharpoonright k)\frac{\chi^{\lambda}(\mu \cup 1^{n-k})}{\dim\simplerep{\lambda}} \end{equation} where $(n \downharpoonright k)$ is the \emph{falling factorial power}, which is defined as ${\fallingfactorial{n}{k} = n(n-1)\dots (n-k+1)}$ for integers $k,n$ with $0 < k \leq n$. Finally, recall that the Jucys-Murphy elements $\{\JM{i}\}_{1 \leq k \leq n} \subseteq \MB{C}[\Sy{n}]$, are defined as \begin{equation*} \JM{1} = 0, \quad\quad \text{and} \quad\quad \JM{k} = (1,k) + (2,k) + \dots + (k-1,k), \;\;\;\;\; 2 \leq k \leq n. \end{equation*} %%%%%%%%%%%%%%%%% The transition measure and co-transition measure %%%%%%%%%%%%%%%%%%%% \subsection{The transition measure and co-transition measure} \label{sect-transition} In this section we review the transition and co-transition measures associated to a Young diagram. Assume that $\lambda \in \partitionsn{n}$ and let $\interlacingx{1}, \dots, \interlacingx{d}$ and $\interlacingy{1}, \dots, \interlacingy{d-1}$ be the interlacing sequences associated to $\lambda$. Recall that $\lambda^{(1)}, \dots, \lambda^{(d)}$ are the partitions of $n+1$ such that $\content{\lambda^{(i)} / \lambda} = \interlacingx{i}$, while $\lambda_{(1)}, \dots, \lambda_{(d-1)}$ are the partitions of $n-1$ such that $\content{\lambda / \lambda_{(i)}} = \interlacingy{i}$. For $\lambda$, the {\emph{transition measure}} $\transition{\lambda}$ and {\emph{co-transition measure} $\cotransition{\lambda}$ on $\MB{R}$ are defined as \begin{equation*} \label{eqn-transition-measure} \transition{\lambda} := \sum_{i=1}^d \frac{\dim(\simplerep{\lambda^{(i)}})}{(n+1)\dim(\simplerep{\lambda})} \delta_{\interlacingx{i}} \quad \text{and} \quad \cotransition{\lambda} := \sum_{i=1}^{d-1} \frac{\dim(\simplerep{\lambda_{(i)}})}{\dim(\simplerep{\lambda})}\delta_{\interlacingy{i}} \end{equation*} respectively, where $\delta_{x}$ is the Dirac delta measure with support on $x \in \MB{R}$. These probability measures were first investigated by Kerov \cite{Ker93}, \cite{Ker00}. They are fundamental tools in the study of the asymptotic representation theory of symmetric groups and its connection to free probability. The $k$th moments associated to $\transition{\lambda}$ and $\cotransition{\lambda}$ are given by \begin{equation*} \moment{k}(\lambda) = \sum_{i = 1}^d \frac{\dim(\simplerep{\lambda^{(i)}})}{(n+1)\dim(\simplerep{\lambda})} \interlacingx{i}^k \quad \text{and} \quad \comoment{k}(\lambda) = \sum_{i = 1}^{d-1} \frac{\dim(\simplerep{\lambda_{(i)}})}{\dim(\simplerep{\lambda})}\interlacingy{i}^k \end{equation*} respectively. {\emph{Boolean cumulants}} linearize convolution of probability measures under the notion of Boolean independence \cite{SW97} and can be defined recursively such that if $\{\Boolean{k}(\lambda)\}_{k \geq 1}$ are the Boolean cumulants associated to $\transition{\lambda}$ then, \begin{equation} \label{eqn-boolean-definition} \sum_{i = 1}^k \moment{k-i}(\lambda)\Boolean{i}(\lambda) = \moment{k}(\lambda). \end{equation} \begin{proposition} \label{prop-boolean-comoment} Let $\lambda \in \partitions$ and $k \geq 0$, then $\Boolean{1}(\lambda) = 0$ and $\Boolean{k+2}(\lambda) = |\lambda|\comoment{k}(\lambda)$. \end{proposition} There is a more algebraic approach to the transition measure due to Biane \cite{B98}. Let $\pr{n-1}: \MB{C}[\Sy{n}] \rightarrow \MB{C}[\Sy{n-1}] \subset \MB{C}[\Sy{n}]$ be the projection map so that for $g \in \Sy{n}$, $\pr{n-1}(g) = g$ if $g \in \Sy{n-1}$ and $0$ otherwise. \begin{proposition} \label{prop-alt-definition-moments} For $\lambda \in \partitionsn{n}$, \begin{equation} \label{eqn-moment} \moment{k}(\lambda) = \norcharrep{\lambda}[\pr{n}(\JM{n+1}^k)] \end{equation} and \begin{equation} \label{eqn-comoments-algebraic} \Boolean{k+2}(\lambda) = |\lambda|\comoment{k}(\lambda) = \norcharrep{\lambda}\Big(\sum_{i=1}^{n} s_i \dots s_{n-1}\JM{n}^ks_{n-1} \dots s_i \Big). \end{equation} \end{proposition} \begin{proof} The statement of \eqref{eqn-moment} appears in \cite[Section 4]{Bia03}. A detailed proof can be found in \cite[Theorem 9.23]{HO07}. \eqref{eqn-comoments-algebraic} follows from the fact that $\norcharrep{\lambda}$ is a class function and from the spectral decomposition of $\JM{n}$ \cite{OkV04}. \end{proof} \cref{prop-alt-definition-moments} is related to the fact that we are working in a noncommutative probability space (that is, a von Neumann algebra equipped with a normal faithful trace). In our case the algebra is $\End(\simplerep{\lambda})\otimes M_{n+1}(\MB{C})$ and $\transition{\lambda}$ then arises from the distribution of a self-adjoint element in this algebra \cite[Proposition 3.3]{B98}. %%%%%%%%%%%%%%%%% Shifted symmetric functions %%%%%%%%%%%%%%%%%%%%%%%% \section{The shifted symmetric functions $\ShiftSym{}$} The {\emph{algebra of shifted symmetric functions}} $\ShiftSym{}$ is a deformation of the classical symmetric functions $\Lambda$. Elements of $\ShiftSym{}$ are ``shifted symmetric'', that is, they become symmetric in the new variables $x_i' = x_i - i$. For a detailed study of $\ShiftSym{}$, see \cite{OO97}. $\ShiftSym{}$ contains shifted analogs of elements from $\Lambda$. These include the {\emph{shifted Schur functions}} $\{s^*_\lambda\}_{\lambda \in \partitions}$ \cite{OO97}, as well as the {\emph{elementary shifted functions}} $\{e^*_k\}_{k \geq 0}$ and {\emph{complete shifted functions}} $\{h^*_k\}_{k \geq 0}$ defined by $\elementaryshift{k} := \shiftschur{(1^k)}$ and $\homogenshift{k} := \shiftschur{(k)}$ respectively. Let $\Symtoshift$ be the linear isomorphism $\Symtoshift: \Lambda \rightarrow \ShiftSym{}$ which sends the classical Schur function $s_\lambda \mapsto s_\lambda^*$. Define the element $\shiftpwr{\lambda} \in \ShiftSym{}$ to then be \begin{equation} \label{eqn-shiftpwr} \shiftpwr{\lambda} := F(p_\lambda), \end{equation} where $p_\lambda$ is the power sum symmetric function in $\Lambda$. The elements $\shiftpwr{\lambda}$ are one of several shifted analogues of the power sums. $\shiftpwr{1},\; \shiftpwr{2},\; \shiftpwr{3} \dots$ are algebraically independent and generate $\ShiftSym{}$ \cite{IO02}. Note that unlike classical power sum symmetric functions, in general $\shiftpwr{\lambda} \neq \shiftpwr{\lambda_1}\shiftpwr{\lambda_2} \dots \shiftpwr{\lambda_r}$ for $\lambda = (\lambda_1,\dots,\lambda_r)$. %%%%%%%%%%%%%%%%% \ShiftSym{} as functions on \partitionsn{} %%%%%%%%%%%%%%%%%% \subsection{$\ShiftSym{}$ as functions on $\partitionsn{}$} \label{sect-shiftsym-as-partition-functions} Let $\funonyd$ be the algebra of functions from $\partitionsn{}$ to $\MB{C}$ with pointwise multiplication. Viewing $\mu = (\mu_1, \dots, \mu_t) \in \partitions$ as the sequence $(\mu_1, \dots, \mu_t, 0, 0, \dots)$, we can evaluate $f \in \ShiftSym{}$ on $\mu$ by setting \begin{equation} \label{eqn-how-to-eval-part-on-shift} f(\mu) = f(\mu_1, \dots, \mu_t, 0, 0, \dots). \end{equation} Since $(\mu_1, \dots, \mu_t, 0, 0, \dots)$ has only a finite number of nonzero values, \eqref{eqn-how-to-eval-part-on-shift} is well-defined. In fact $f$ is uniquely defined by its values on $\partitionsn{}$. Thus $\ShiftSym{}$ may be realized as a subalgebra of $\funonyd$ \cite{KO94}, \cite{OO97}. \begin{proposition} \label{prop-value-prwshift} \cite{OO97} For $\mu \in \partitionsn{k}$, $\lambda \in \partitionsn{n}$, \begin{equation} \shiftpwr{\mu}(\lambda) = \begin{cases} \frac{(n \downharpoonright k)}{\dim \simplerep{\lambda}}\charrep{\lambda}(\mu\cup 1^{n-k}) & k \leq n\\ 0 & \text{otherwise.}\\ \end{cases} \end{equation} \end{proposition} \begin{remark} We will later use the fact that $\shiftpwr{1} = x_1 + x_2 + \dots$, so that $\shiftpwr{1}(\lambda) = |\lambda|$ for all $\lambda \in \partitions$. \end{remark} In \cref{sect-transition} we introduced the moments $\{\moment{k}(\lambda)\}$ (respectively $\{\comoment{k}(\lambda)\}$) of the transition measure (respectively co-transition measure) associated to a partition $\lambda$ and the corresponding Boolean cumulants $\{\Boolean{k}(\lambda)\}$\omitt{(respectively $\{\coboolean{k}\}$)}. We can interpret all of these as elements of $\funonyd$ via \begin{equation*} \lambda \xmapsto{\moment{k}} \moment{k}(\lambda), \quad \lambda \xmapsto{\comoment{k}} \comoment{k}(\lambda), \quad \text{and} \quad \lambda \xmapsto{ \Boolean{k} } \Boolean{k}(\lambda). \end{equation*} \begin{proposition} \cite[Theorem 6.4]{L09} \label{prop-moments-as-sym} As elements of $\funonyd$, $\moment{k}$ and $\Boolean{k}$ belong to $\ShiftSym{}$. \end{proposition} \begin{remark} \label{remark-moments-as-sym} In \cite{L09} Section 5, Lassalle shows that with the appropriate alphabet $A_\lambda$ (which is specific to each partition $\lambda$), $\moment{k}(\lambda) = h_k(A_\lambda)$ and $\Boolean{k}(\lambda) = (-1)^{k-1}e_k(A_\lambda)$. \end{remark} %%%%%%%%%%%%%%%%% The Heisenberg category $\Heisencat$ %%%%%%%%%%%%%%%%%%%% \section{The algebra $\Hcenter$} In \cite{Kho14}, Khovanov defined an additive $\MB{C}$-linear monoidal category $\Heisencat$ which we will call the {\emph{Heisenberg category}}. The unit object in $\Heisencat$ is denoted by $\UnitModule$. In this paper we study the endomorphism algebra $\Hcenter$. $\Hcenter$ is a $\MB{C}$-algebra generated by planar diagrams modulo local relations. The diagrams are closed oriented compact 1-manifolds immersed in the strip $\MB{R} \times [0,1]$, modulo isotopy. Multiplication corresponds to juxtaposition of diagrams. The local relations are: \vspace{6mm} \begin{equation} \label{up down double crossings} \begin{tikzpicture}[baseline=(current bounding box.center),scale =.75] % double cross up down \draw[thick,->] (0,0) .. controls (1,1) .. (0,2); \draw[thick,->] (1,2) .. controls (0,1) .. (1,0); \node at (2,1) {$=$}; \draw[thick,->] (3,0) -- (3,2); \draw[thick,->] (4,2) -- (4,0); % double cross down up \draw[thick,->] (6,2) .. controls (7,1) .. (6,0); \draw[thick,->] (7,0) .. controls (6,1) .. (7,2); \node at (8,1) {$=$}; \draw[thick,->] (9,2) -- (9,0); \draw[thick,->] (10,0) -- (10,2); \node at (11,1) {$-$}; \draw[thick,->] (12,2) arc (180:360:0.75); \draw[thick,->] (13.5,0) arc (0:180:0.75); \end{tikzpicture} \end{equation} \vspace{4mm} \begin{equation} \label{anti-clockwise and left curl} \begin{tikzpicture}[baseline=(current bounding box.center), scale = .85] % anti-clockwise circle \draw[<-,thick] (0,1) arc (180:-180:.5); \node at (1.5,1) {$= 1$}; % left curl \draw[thick] (4,1) .. controls (4,1.5) and (4.7,1.5) .. (4.9,1); \draw[thick] (4,1) .. controls (4,0.5) and (4.7,0.5) .. (4.9,1); \draw[thick] (5,0) .. controls (5,0.5) .. (4.9,1) ; \draw[thick,->] (4.9,1) .. controls (5,1.5) .. (5,2); \node at (5.5,1) {$= 0$}; \end{tikzpicture} \end{equation} \vspace{4mm} \begin{equation} \label{eqn-symmetric-group-relations} \begin{tikzpicture}[baseline=(current bounding box.center),scale = .75] % double cross up up \draw[->,thick] (0,0) .. controls (1,1) .. (0,2); \draw[->,thick] (1,0) .. controls (0,1) .. (1,2); \node at (2,1) {$=$}; \draw[->,thick] (3,0) -- (3,2); \draw[->,thick] (4,0) -- (4,2); % triple \draw[->,thick] (6,0) -- (8,2); \draw[->,thick] (8,0) -- (6,2); \draw[->,thick] (7,0) .. controls (8,1) .. (7,2); \node at (9,1) {$=$}; \draw[->,thick] (10,0) -- (12,2); \draw[->,thick] (12,0) -- (10,2); \draw[->,thick] (11,0) .. controls (10,1) .. (11,2); \node at (12.3,0) {.}; \end{tikzpicture} \end{equation} \vspace{6mm} The relations \eqref{up down double crossings}-\eqref{anti-clockwise and left curl} are motivated by the Heisenberg relation $pq = qp + 1$, where $p$ and $q$ are the two generators of the Heisenberg algebra, while the relations \eqref{eqn-symmetric-group-relations} are motivated by the symmetric group relations. It is convenient to denote a right curl by a dot on a strand, and a sequence of $d$ right curls by a dot with a $d$ next to it: \begin{center} \begin{tikzpicture}[scale = .94] \draw[thick,->] (0,0) -- (0,2); \draw (0,1) node {\textbullet}; \draw (1,1) node {$:=$}; \draw[thick] (3,1) .. controls (3,1.5) and (2.3,1.5) .. (2.1,1); \draw[thick] (3,1) .. controls (3,0.5) and (2.3,0.5) .. (2.1,1); \draw[thick] (2,0) .. controls (2,0.5) .. (2.1,1); \draw[thick] (2.1,1) .. controls (2,1.5) .. (2,2) [->]; \node at (3.4,0) {,}; \draw[->,thick] (5,0) -- (5,2); \draw (5,1) node {\textbullet}; \draw (4.7,1) node {$d$}; \draw (6,1) node {$:=$}; \draw[->,thick] (7,0) -- (7,2); \draw (7,0.4) node {\textbullet}; \draw (7,0.8) node {\textbullet}; \draw (7,1.2) node {\textbullet}; \draw (7,1.6) node {\textbullet}; \draw (8.4,1) node {$d$ dots}; \draw [decorate,decoration={brace,amplitude=8pt},xshift=-4pt,yshift=0pt] (7.4,1.7) -- (7.4,.3) node [black,midway,xshift=0cm,yshift = .6cm] {}; \node at (8,0) {.}; \end{tikzpicture} \end{center} The relations \eqref{eqn-symmetric-group-relations} allow us to identify elements of $\MB{C}[\Sy{n}]$ with linear combinations of diagrams with $n$ upward oriented strands. For $x \in \MB{C}[\Sy{n}]$ our notation for such a linear combination of diagrams is a box with an $x$ in it \begin{center} \label{picture-sym-group-emedding} \begin{tikzpicture}[scale = .94] %\node at (-3.5,0) {$\SymtoHplus{n}(x) \;\; =: \;\;$}; \draw[thick,->] (-1.7,-1)--(-1.7,1); \draw[thick,->] (-.8,-1)--(-.8,1); \draw[thick,->] (-2,-1)--(-2,1); %\node at (-1.7,.6) {\arrowlines}; %\node at (-.8,.6) {\arrowlines}; \node at (-1.4,.7) {$\cdot$}; \node at (-1.2,.7) {$\cdot$}; \node at (-1.0,.7) {$\cdot$}; \draw[fill=white,thick] (-2.3,-.3) rectangle (-.5,.3); \node at (-1.3,0) {\scriptsize{$x$}}; \node at (-.3,-.9) {.}; \end{tikzpicture} \end{center} \newpage Next set \begin{center} \begin{tikzpicture} \node at (-1.1,2) {$c_k$}; \node at (.2,2) {$:=$}; \draw[thick] (2,2) circle (.5cm); \draw[thick] (1.5,2) -- (1.6,1.9); \draw[thick] (1.5,2) -- (1.4,1.9); \draw[fill=black] (1.65,2.35) circle (.08cm); \node at (1.4,2.7) {$k$}; \node at (4,2) {and}; \node at (7.5,2.3) {\begin{tikzpicture} \node at (6.3,2) {$\ctilde{k} \;\;\; := \;\;\;$}; \draw[thick] (8,2) circle (.5cm); \draw[thick] (7.5,2) -- (7.6,2.1); \draw[thick] (7.5,2) -- (7.4,2.1); \draw[fill=black] (7.65,2.35) circle (.08cm); \node at (7.3,2.7) {$k$}; \node at (8.8,1.6) {$.$}; \end{tikzpicture}}; \end{tikzpicture} \end{center} \begin{theorem} \label{Thm-End-Iso} \cite[Prop. 3]{Kho14} There are algebra isomorphisms \begin{equation} \Hcenter \cong \MB{C}[c_0, c_1, \dots ] \cong \MB{C}[\ctilde{2},\ctilde{3},\dots]. \end{equation} \end{theorem} Note that it follows from the relations in \eqref{anti-clockwise and left curl} that $\ctilde{0} = 1$ and $\ctilde{1} = 0$. \begin{lemma} \cite[Prop. 2]{Kho14} \label{lemma-c-ctilde-rln} For $k > 0$, \begin{equation} \label{anticlockwise to clockwise} \ctilde{k+1} = \sum_{i = 0}^{k-1} \tilde{c}_i c_{k-1-i}. \end{equation} \end{lemma} Let $E_\lambda$ be the Young idempotent associated with $\lambda$ so that $\MB{C}[\Sy{n}]E_\lambda \cong \simplerep{\lambda}$. Also let $\sigma_\lambda \in \Sy{n}$ be an element of cycle type $\lambda$ and set \begin{center} \begin{tikzpicture} \node at (0,0) {\begin{tikzpicture}[scale =.73] \node at (-4,0) {$\alpha_\lambda \quad := $}; \draw[thick] (0,0) circle (1.8 cm); \draw[thick] (0,0) circle (1.6 cm); \draw[thick] (0,0) circle (1 cm); \draw[fill=white,thick] (-2,-.3) rectangle (-.5,.3); \node at (-1,.5) {$\cdot$}; \node at (-1.15,.55) {$\cdot$}; \node at (-1.3,.6) {$\cdot$}; \node at (-1.3,0) {$\sigma_\lambda$}; \node[rotate = 180] at (1.8,0) {\arrowlines}; \node[rotate = 180] at (1.6,0) {\arrowlines}; \node[rotate = 180] at (1,0) {\arrowlines}; \end{tikzpicture}}; \node at (3,-1.1) {,}; \node at (8,0) {\begin{tikzpicture}[scale = .73] \node at (-5,0) {$\tilde{E}_\lambda \quad := $}; \node at (-3,-.3) {\small{$\dim \simplerep{\lambda}$}}; \node at (-3,.3) {$1$}; \draw[thick] (-3.8,0) -- (-2.3,0); %circles \draw[thick] (0,0) circle (1.8 cm); \draw[thick] (0,0) circle (1.6 cm); \draw[thick] (0,0) circle (1 cm); \draw[fill=white,thick] (-2,-.3) rectangle (-.5,.3); \node at (-1,.5) {$\cdot$}; \node at (-1.15,.55) {$\cdot$}; \node at (-1.3,.6) {$\cdot$}; \node at (-1.3,0) {\small{$\Youngidempotent{\lambda}$}}; \node[rotate = 180] at (1.8,0) {\arrowlines}; \node[rotate = 180] at (1.6,0) {\arrowlines}; \node[rotate = 180] at (1,0) {\arrowlines}; \end{tikzpicture}}; \node at (11,-1.2) {.}; \end{tikzpicture} \end{center} Because the diagrams are closed, the local relations imply that all choices of $\sigma_\lambda$ give the same element of $\Hcenter$, so $\alpha_\lambda$ is well-defined. We write $\alpha_k := \alpha_{(k)}$. \begin{proposition} \label{cor-end-generators} The elements $\alpha_1, \alpha_2, \dots$ are algebraically independent generators of $\Hcenter$. \end{proposition} For each $n \geq 0$, Khovanov defines a functor $\htobimod{n}: \Heisencat \rightarrow \bimodcatn{n}$, where $\bimodcatn{n}$ is a bimodule category for symmetric groups whose objects are all right $\MB{C}[\Sy{n}]$-modules (see \cite{Kho14} for details). When restricted to $\Hcenter$, $\htobimod{n}$ can be interpreted as a surjective homomorphism into $Z(\MB{C}[\Sy{n}])$. Below we give the value of $\htobimod{n}$ on $c_k$, $\ctilde{k}$, and $\alpha_k$ in $\Hcenter$. \begin{lemma} \label{lemma-what-is-c_k-in-center} If $n \geq 1$, then \begin{enumerate} \setlength\itemsep{-.3em} \item \label{eqn-ck-value} $ \displaystyle \htobimod{n}(c_k) = \sum_{i = 1}^n s_i \cdots s_{n-1} \JM{n}^k s_{n-1} \cdots s_i,$ \item \label{eqn-cktilde-value} $\htobimod{n}(\tilde{c}_k) = \pr{n}(\JM{n+1}^k).$ \item \label{eqn-alpha-value} $\htobimod{n}(\alpha_\mu) = \begin{cases} \classsum{\mu}{n} & \text{if $|\mu| \leq n$}\\ 0 & \text{otherwise}. \end{cases}$ \end{enumerate} \end{lemma} %%%%%%%%%% \Hcenter as functions on \partitionsn{} %%%%%%%%%%%%%% \section{The isomorphism $\primaryiso: \Hcenter \longrightarrow \ShiftSym{}$} In this section we establish the algebra isomorphism $\Hcenter \cong \ShiftSym{}$. The proof is somewhat analogous to a proof of Ivanov and Kerov \cite[Theorem 9.1]{IK99}. For any $\lambda \in \partitionsn{n}$, composing $\htobimod{n}$ with the normalized character $\norcharrep{\lambda}$ gives a map \begin{equation*} (\norcharrep{\lambda} \circ \htobimod{n}): \Hcenter \rightarrow \MB{C} \end{equation*} and allows us to define a homomorphism $\primaryiso: \Hcenter \rightarrow \funonyd$. For $x \in \Hcenter$, \begin{equation*} [\primaryiso (x)](\lambda) := (\norcharrep{\lambda} \circ \htobimod{n})(x). \end{equation*} Combining part~\ref{eqn-alpha-value} of \cref{lemma-what-is-c_k-in-center} with equation \eqref{prop-normalized-char-class-sum} implies that for $\mu \in \partitionsn{k}$ \begin{equation} \label{eqn-value-alpha} [\primaryiso(\alpha_\mu)](\lambda) = \begin{cases} \frac{(n \downharpoonright k)}{\dim\simplerep{\lambda}}\chi^{\lambda}(\mu \cup 1^{n-k}) & \text{if $k \leq n$}\\ 0 & \text{otherwise.} \end{cases} \end{equation} \begin{theorem} \label{thm-main-1} The map $\primaryiso$ induces an algebra isomorphism $\Hcenter \rightarrow \ShiftSym{} \subseteq \funonyd$ with $\alpha_\mu \xmapsto{\primaryiso} \shiftpwr{\mu}$. \end{theorem} \begin{proof} Let $\lambda \in \partitionsn{n}$. $\primaryiso$ is an algebra homomorphism because $\htobimod{n}$ is a homomorphism from $\Hcenter$ to $Z(\MB{C}[\Sy{n}])$ and $\norcharrep{\lambda}$ is a homomorphism when restricted to $Z(\MB{C}[\Sy{n}])$. By \cref{prop-value-prwshift} and \eqref{eqn-value-alpha}, $\alpha_\mu$ maps to $\shiftpwr{\mu}$. Since the $\{\shiftpwr{k}\}_{k \geq 1}$ (respectively $\{\alpha_k\}_{k \geq 1}$) are algebraically independent generators of $\ShiftSym{}$ (respectively $\Hcenter$), $\primaryiso$ must be an isomorphism. \end{proof} \begin{table}[h] \begin{center} {\tabulinesep=1.2mm \begin{tabular}{| m{2.4cm} | m{5cm} |} \hline $ \ShiftSym{}$ & diagram in $\Hcenter$ \\ \hline \hline $\shiftpwr{\lambda}$ & \begin{tikzpicture}[scale = .55] \node at (-2.5,0) {}; \node at (0,2) {}; \draw[thick] (0,0) circle (1.85 cm); \draw[thick] (0,0) circle (1.6 cm); \draw[thick] (0,0) circle (1 cm); \draw[fill=white,thick] (-2,-.33) rectangle (-.5,.33); \node at (-1,.5) {$\cdot$}; \node at (-1.15,.55) {$\cdot$}; \node at (-1.3,.6) {$\cdot$}; \node at (-1.3,0) {\small{$\sigma_\lambda$}}; \node[rotate = 180] at (1,0) {\arrowlines}; \node[rotate = 180] at (1.6,0) {\arrowlines}; \node[rotate = 180] at (1.85,0) {\arrowlines}; \node at (-3.6,0) {}; \end{tikzpicture} \\ \hline $\shiftschur{\lambda}$ & \begin{tikzpicture}[scale = .55] \node at (-2.5,0) {}; \node at (0,2) {}; \draw[thick] (0,0) circle (1.85 cm); \draw[thick] (0,0) circle (1.6 cm); \draw[thick] (0,0) circle (1 cm); \node at (0,-1.7) {}; \draw[fill=white,thick] (-2,-.33) rectangle (-.5,.33); \node at (-1,.5) {$\cdot$}; \node at (-1.15,.55) {$\cdot$}; \node at (-1.3,.6) {$\cdot$}; \node at (-3,-.3) {\scriptsize{$\dim \simplerep{\lambda}$}}; \node at (-3,.35) {\small{$1$}}; \node at (-1.3,0) {\scriptsize{\text{$\idempotent{\lambda}$}}}; \draw[thick] (-3.8,0) -- (-2.3,0); \node[rotate = 180] at (1,0) {\arrowlines}; \node[rotate = 180] at (1.6,0) {\arrowlines}; \node[rotate = 180] at (1.85,0) {\arrowlines}; \end{tikzpicture} \\ \hline $\homogenshift{k}$ & \begin{tikzpicture}[scale = .55] \node at (-2.5,0) {}; \node at (0,2) {}; \draw[thick] (0,0) circle (1.85 cm); \draw[thick] (0,0) circle (1.6 cm); \draw[thick] (0,0) circle (1 cm); \draw[fill=white,thick] (-2,-.35) rectangle (-.5,.35); %\node at (-2.7,0) {$k!$}; \node at (-1,.5) {$\cdot$}; \node at (-1.15,.55) {$\cdot$}; \node at (-1.3,.6) {$\cdot$}; \node at (-1.3,0) {\scriptsize{\text{$\idempotent{(k)}$}}}; \node[rotate = 180] at (1,0) {\arrowlines}; \node[rotate = 180] at (1.6,0) {\arrowlines}; \node[rotate = 180] at (1.85,0) {\arrowlines}; \node at (-3.6,0) {}; \end{tikzpicture} \\ \hline $\elementaryshift{k}$ & \begin{tikzpicture}[scale = .55] \node at (-2.5,0) {}; \node at (0,2) {}; \draw[thick] (0,0) circle (1.85 cm); \draw[thick] (0,0) circle (1.6 cm); \draw[thick] (0,0) circle (1 cm); %\node at (-2.7,0) {$k!$}; \draw[fill=white,thick] (-2,-.37) rectangle (-.5,.37); \node at (-1,.5) {$\cdot$}; \node at (-1.15,.55) {$\cdot$}; \node at (-1.3,.6) {$\cdot$}; \node at (-1.3,0) {\scriptsize{\text{$\idempotent{(1^k)}$}}}; \node[rotate = 180] at (1,0) {\arrowlines}; \node[rotate = 180] at (1.6,0) {\arrowlines}; \node[rotate = 180] at (1.85,0) {\arrowlines}; \node at (-3.6,0) {}; \end{tikzpicture} \\ \hline $\moment{k}$ & \quad\quad\quad\cktildepicture \\ \hline $\Boolean{k+2} = \shiftpwr{1}\comoment{k}$ & \quad\quad\quad\ckpicture \\ \hline \end{tabular}} \end{center} \caption{\label{i-class-table} A dictionary between $\ShiftSym{}$ and diagrams in $\Hcenter$.} \label{node-classification} \end{table} \begin{corollary} \label{cor-Schur} The isomorphism $\primaryiso$ sends $\tilde{E}_\lambda \xmapsto{\primaryiso} s^*_\lambda$. \end{corollary} \cref{thm-main-1} and \cref{cor-Schur} give graphical realizations of some important bases of $\ShiftSym{}$. Now we go the other way, and describe Khovanov's curl generators $\ctilde{k}$ and $c_k$ as elements of $\ShiftSym{}$. It is this description that makes an explicit connection between $\Heisencat$ and the transition and co-transition measures of Kerov. \begin{theorem} \label{thm-moment-images} The isomorphism $\primaryiso$ sends: \begin{enumerate} \item $\ctilde{k} \mapsto \moment{k} \in \ShiftSym{}$, \item $c_k \mapsto \shiftpwr{1}\comoment{k} = \Boolean{k+2} \in \ShiftSym{}$. \end{enumerate} \end{theorem} \begin{proof} This follows from \cref{prop-alt-definition-moments} and \cref{lemma-what-is-c_k-in-center}. \end{proof} \begin{remark} \label{remark-curl-recursive-relations} \cref{thm-moment-images} and \cref{remark-moments-as-sym} together imply that the recursive relationships for $\{\moment{k}\}$ and $\{\Boolean{k}\}$ and the recursive relationships for $\{c_k\}$ and $\{\ctilde{k}\}$ in \cref{lemma-c-ctilde-rln} are both consequences of the well-known relationship between the elementary and homogeneous symmetric functions: \begin{equation*} \sum_{i = 0}^k (-1)^i e_ih_{n-i} = 0. \end{equation*} \end{remark} \newpage \begin{example} In $\ShiftSym{}$ we have $\shiftpwr{(2)}\shiftpwr{(2)} = \shiftpwr{(2,2)} + 4\shiftpwr{(3)} + 2\shiftpwr{(1,1)}$. In $\Hcenter$ this appears as \begin{center} \begin{tikzpicture}[scale = .8] %left side %top \draw[thick] (0,0) arc (0:180:8mm); \draw[thick] (-.28,0) arc (0:180:5mm); \draw[thick] (2,0) arc (0:180:8mm); \draw[thick] (1.71,0) arc (0:180:5mm); %middle \draw[thick] (-1.6,0) to [in = 90, out = 270] (-1.275,-.35); \draw[thick] (-1.6,-.35) to [in = 270, out = 90] (-1.275,0); \draw[thick] (0,0)--(0,-.35); \draw[thick] (-.28,0)--(-.28,-.35); \draw[thick] (.4,0) to [in = 90, out = 270] (.71,-.32); \draw[thick] (.4,-.32) to [in = 270, out = 90] (.71,0); \draw[thick] (2,0)--(2,-.35); \draw[thick] (1.71,0)--(1.71,-.35); %bottom \draw[thick] (0,-.3) arc (180:0:-8mm); \draw[thick] (-.28,-.3) arc (180:0:-5mm); \draw[thick] (2,-.3) arc (180:0:-8mm); \draw[thick] (1.71,-.3) arc (180:0:-5mm); %equality \node at (2.8,-.1) {$=$}; % (2,2) terms %top \draw[thick] (6,0) arc (0:180:13mm); \draw[thick] (5.7,0) arc (0:180:10mm); \draw[thick] (5.4,0) arc (0:180:7mm); \draw[thick] (5.1,0) arc (0:180:4mm); %middle \draw[thick] (3.4,0) to [in = 90, out = 270] (3.7,-.35); \draw[thick] (3.4,-.35) to [in = 270, out = 90] (3.7,0); \draw[thick] (4,0) to [in = 90, out = 270] (4.3,-.35); \draw[thick] (4,-.35) to [in = 270, out = 90] (4.3,0); \draw[thick] (6,0)--(6,-.3); \draw[thick] (5.7,0)--(5.7,-.3); \draw[thick] (5.4,0)--(5.4,-.3); \draw[thick] (5.1,0)--(5.1,-.3); %bottom \draw[thick] (6,-.3) arc (180:0:-13mm); \draw[thick] (5.7,-.3) arc (180:0:-10mm); \draw[thick] (5.4,-.3) arc (180:0:-7mm); \draw[thick] (5.1,-.3) arc (180:0:-4mm); %plus \node at (6.7,0) {\Large{$+$}}; %number \node at (7.5,0) {4}; %top \draw[thick] (10,0) arc (0:180:10mm); \draw[thick] (9.7,0) arc (0:180:7mm); \draw[thick] (9.4,0) arc (0:180:4mm); %middle \draw[thick] (8,0) to [out = 270, in = 90] (8.6,-.3); \draw[thick] (8.3,0) to [out = 270, in = 90] (8,-.3); \draw[thick] (8.6,0) to [out = 270, in = 90] (8.3,-.3); \draw[thick] (10,0) -- (10,-.3); \draw[thick] (9.7,0) -- (9.7,-.3); \draw[thick] (9.4,0) -- (9.4,-.3); %bottom \draw[thick] (10,-.3) arc (180:0:-10mm); \draw[thick] (9.7,-.3) arc (180:0:-7mm); \draw[thick] (9.4,-.3) arc (180:0:-4mm); %plus \node at (10.8,0) {\Large{$+$}}; %number \node at (11.6,0) {$2$}; %top \draw[thick] (13.4,0) arc (0:180:7mm); \draw[thick] (13.1,0) arc (0:180:4mm); %middle \draw[thick] (13.4,0) -- (13.4,-.3); \draw[thick] (13.1,0) -- (13.1,-.3); \draw[thick] (12.3,0) -- (12.3,-.3); \draw[thick] (12,0) -- (12,-.3); %bottom \draw[thick] (13.4,-.3) arc (180:0:-7mm); \draw[thick] (13.1,-.3) arc (180:0:-4mm); %arrow \node[rotate = 90] at (-.29,0) {\arrow}; \node[rotate = 90] at (0,0) {\arrow}; \node[rotate = 90] at (1.71,0) {\arrow}; \node[rotate = 90] at (2,0) {\arrow}; \node[rotate = 90] at (6,0) {\arrow}; \node[rotate = 90] at (5.7,0) {\arrow}; \node[rotate = 90] at (5.4,0) {\arrow}; \node[rotate = 90] at (5.1,0) {\arrow}; \node[rotate = 90] at (10,0) {\arrow}; \node[rotate = 90] at (9.7,0) {\arrow}; \node[rotate = 90] at (9.4,0) {\arrow}; \node[rotate = 90] at (13.4,0) {\arrow}; \node[rotate = 90] at (13.1,0) {\arrow}; %\node at (13.5,-1) {.}; %\draw[fill = black] (0,0)--(.15,0)--(.075,.1)--(0,0); \end{tikzpicture} \end{center} and can be computed independently via the local relations. \end{example} In \cite{Kho14}, Khovanov introduced three involutive autoequivalences on $\Heisencat$. Only one of these, which we denote as $\involution$, acts non-trivially on $\Hcenter$ where it gives an involutive algebra automorphism. For diagram $D \in \Hcenter$, we have $\involution(D) := (-1)^{\crossingplusdots(D)}D$ where $\crossingplusdots(D)$ is the total number of dots and crossings in the diagram. In Section 4 of \cite{OO97}, Okounkov and Olshanski identified an involutive algebra automorphism $\shiftinvolution:\ShiftSym{} \rightarrow \ShiftSym{}$ such that for $f \in \ShiftSym{}$ and $\lambda \in \partitions$, $[\shiftinvolution(f)](\lambda) = f(\lambda')$ where $\lambda'$ is the conjugate partition to $\lambda$. \begin{proposition} The involution $\involution$ on $\Hcenter$ coincides with the involution $\shiftinvolution$ on $\ShiftSym{}$. \end{proposition} \acknowledgements{The authors would like to thank Ben Elias, Alexander Ellis, Sara Billey, Eugene Gorsky, Aaron Lauda, Carson Rogers, and Alistair Savage for helpful conversations. We would also like to thank Monica Vazirani for her valuable comments after a careful reading of an earlier draft of this paper. H.K. would like to thank Mikhail Khovanov for his suggestion to look for a relationship between the Heisenberg category and the combinatorics of symmetric functions.} \printbibliography \end{document}