\ProvidesPackage{spinHeisenCenterMACROS}[2016/04/20 v1.0 My own macros] %%%%%%%%%%%% Some General MACROS %%%%%%%%%%%%%%%%%%%%% \newcommand{\F}{\mathbb{F}} %%% field \newcommand{\Z}{\mathbb{Z}} %%% field \newcommand{\N}{\mathbb{Z}_{\geq 0}} %%% field \newcommand{\Q}{\mathbb{Q}} %%% field \newcommand{\C}{\mathbb{C}} %%% field \newcommand{\wh}{\widehat} %%% Short for \widehat %%% \newcommand{\MC}[1]{\mathcal{#1}} %%% Shortcut for \mathcal %%% \newcommand{\B}{\pmb} %%% Short for \pmb %%% \newcommand{\Fr}{\mathfrak} %%% Short for \mathfrak %%% \newcommand{\MB}{\mathbb} %%% Short for \mathbb %%% \newcommand{\p}{\varphi} %%% Short for \varphi %%% \DeclareMathOperator{\Hom}{Hom} %%% Hom with capital "H" %%% \DeclareMathOperator{\ind}{Ind} %%% Ind %%% \DeclareMathOperator{\coind}{coInd} %%% coInd %%% \DeclareMathOperator{\res}{Res} %%% Res %%% \DeclareMathOperator{\cont}{Cont} %%% Cont %%% \DeclareMathOperator{\spec}{Spec} %%% Spec %%% \DeclareMathOperator{\soc}{soc} %%% soc %%% \DeclareMathOperator{\cosoc}{cosoc} %%% cosoc %%% \DeclareMathOperator{\im}{Im} %%% Im %%% \DeclareMathOperator{\Dim}{dim} %%% dim %%% \DeclareMathOperator{\tr}{Tr} %%% Tr %%% \DeclareMathOperator{\ad}{ad} %%% ad %%% \DeclareMathOperator{\End}{End} %%% End %%% \DeclareMathOperator{\id}{id} %%% id %%% \DeclareMathOperator{\Mod}{\dash \text{mod}} %%% mod %%% \DeclareMathOperator{\wt}{wt} %%% wt %%% \DeclareMathOperator{\ext}{Ext} %%% Ext %%% \DeclareMathOperator{\HOM}{HOM} %%% HOM %%% \DeclareMathOperator{\dash}{-} %%% dash %%% \DeclareMathOperator{\UnitModule}{\mathbbm{1}} %%% Unit Module %%% \newcommand{\zero}{\mathbf{0}} %%% Zero Module %%% \newcommand{\partitions}{\MC{P}} %%% partitions, \mathcal{P} %%% \newcommand{\partitionsn}[1]{\MC{P}_{#1}} %%% partitions of specific x, \mathcal{P}_x, for x variable %%% \newcommand{\npartitions}{\MC{NP}} %%% noncrossing partitions, \mathcal{P}_n %%% \newcommand{\npartitionsn}[1]{\MC{NP}_{#1}} %%% noncrossing partitions for supecific x, \mathcal{P}_x, for x variable %%% \newcommand{\bimod}{\text{- bimod}} %%% - bimod %%% \newcommand{\shape}[1]{\text{sh}(#1)} %%% sh(x), for x variable %%% \newcommand{\ydcell}{\;\framebox(7,7){}\;} %%% Makes cell for Young diagram %%% \newcommand{\Karoubi}[1]{\text{Kar}(#1)} %%% Karoubi envelope, Kar(x) for x a variable %%% %%%%%%%%%%% MACROS for non-spin Heisenberg category project %%%%%%%%%%%%%%%%% \newcommand{\Sy}[1]{\ensuremath{S_{#1}}} %%% Symmetric group, \mathcal{S}_{x} where x is variable %%% \newcommand{\ShiftSym}[1][]{\Lambda^*_{#1}} %%% Shifted symmetric polynomials, \Lambda^*_x for x variable %%% \newcommand{\GrothG}[1]{G_0(#1)} %%% Grothendieck group, simples, G_0(x) for x variable %%% \newcommand{\GrothK}[1]{K_0(#1)} %%% Grothendieck group, projectives, K_0(x) for x variable %%% \newcommand{\speclcos}[2]{\MC{LC}^{#1}_{#2}} %%% Family of left coset representatives, \MC{LC}^x_y for x,y variables %% \newcommand{\specrcos}[2]{\MC{RC}^{#1}_{#2}} %%% Family of right coset representatives \MC{RC}^x_y for x,y variables %% \newcommand{\partitioncycle}[1]{\pi_{#1}} %%% Conjugacy class rep, \rho_{x} for x variable %% \newcommand{\partitioncyclen}[2]{\sigma_{#1,#2}} %%% Another conjugacy class rep, \rho_{x,y} for x, y variable %%% \newcommand{\longestelement}[1]{w_{0,#1}} %%% Longest element in symmetric group, \omega_{0,x} for x variable %% \newcommand{\simplerep}[1]{L^{#1}} %%% Symmetric group simple rep, V^x for x variable %%% \newcommand{\charrep}[1]{\chi^{#1}} %%% Simple rep character, \chi^x for x variable %%% \newcommand{\classsum}[2]{A_{#1,#2}} %%% Character element, a_{x,y} for x,y variables %%% \newcommand{\partembedding}[2]{\phi_{#1,#2}} %%% Embedding for partitions, \phi_{x,y} for x, y variables %%% \newcommand{\symembedding}[2]{\iota_{#1,#2}} %%% Embedding for symmetric groups, \iota_{x,y} for x,y variables %%% \newcommand{\bimodcat}{\MC{S}'} %%% Symmetric group bimodule category \MC{S}' %%% \newcommand{\bimodcatn}[1]{\MC{S}_{#1}'} %%% Symmetric group bimodule category \MC{S}_n' %%% \newcommand{\htobimod}[1]{f_{#1}^{\Heisencat}} %%% Functor from H' to S'_n, \MC{F}_n %%% \newcommand{\norcharrep}[1]{\widetilde{\chi}^{#1}} %%% Normalized character, \wh{\chi}^{x} for x variable %%% \newcommand{\funonyd}{\text{Fun}(\partitions, \MB{C})} %%% Functions on YD, Func_Y %%% \newcommand{\funonsp}{\text{Fun}(\strictpartitions{}, \MB{C})} %%% Functions on SP %%% \newcommand{\assocgraded}{\text{gr}} %%% Associated graded, gr %%% \newcommand{\shiftpwr}[1]{p^{\#}_{#1}} %%% Shifted power functions for normalized character, p^#_x for variable x %%% \newcommand{\shiftschur}[1]{s^*_{#1}} %%% Shifted Schur functions, s^*_x for variable x %%% \newcommand{\primaryiso}{\p} %%% Isomorphism from End(1) to shifted symmetric functions, \p %%% \newcommand{\congclasssum}[2]{C_{#1,#2}} %%% Conjugacy class sun, C_{x,y} for x,y variables %%% \newcommand{\loworderterms}{\text{l.o.t.}} %%% Lower order terms, l.o.t %%% \newcommand{\nounity}[1]{\overline{#1}} %%% Partition without unity parts, \overline{x} for x variable %%% \newcommand{\idempotent}[1]{E_{#1}} %%% idempotent associated to x, e(x) for x variable %%% \newcommand{\ctilde}[1]{\tilde{c}_{#1}} %%% Counter-clockwise circle with x dots, \tilde{c}_x, for x variable %%% \newcommand{\Hcenter}{\End_{\MC{H}'}(\UnitModule)} %%% The center of H', 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{\uptransitionprob}[2]{\wh{q}_{#1}(#2)} %%% Transition probability, p_x(y) for x,y variables %%% \newcommand{\downtransitionprob}[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 moments, \widecheck{\sigma_x} for x variable %%% \newcommand{\tranmomentseries}[1]{\MC{\wh{M}}_{#1}(z)} %%% Transititon moment generating function, \MC{\wh{M}}(z) %%% \newcommand{\cotranmomentseries}[1]{\MC{\widecheck{M}}_{#1}(z)} %%% Co-transition moment generating function, \MC{\widecheck{M}}(z) %%% \newcommand{\Boolean}[1]{\wh{b}_{#1}} %%% Boolean cumulant for transition, \wh{B}_x for x,y variables %%% \newcommand{\coboolean}[1]{\widecheck{B}_{#1}} %%% Boolean cumulant for co-transition, \widecheck{B}_x for x,y variables %%% \newcommand{\booleanseries}[1]{\MC{\wh{B}}_{#1}(z)} %%% Boolean generating series, \MC{B}(z) %%% \newcommand{\elementaryshift}[1]{e^*_{#1}} %%% Shifted elementary symmetric function, e^*_x for x variable %%% \newcommand{\homogenshift}[1]{h^*_{#1}} %%% Shifted homogeneous symmetric function, 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]{\mathrm{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{\conj}{\mathrm{Conj}} %%% Conjugacy class of pair of partitions \newcommand{\conjClass}[2]{\mathrm{Conj}_{#1,#2}} %%% Conjugacy class of pair of partitions \newcommand{\conjRep}{\mathrm{ConjRep}} %%% Conjugacy class representatives \newcommand{\conjOdd}{\mathrm{Conj}_{odd}} %%% Odd partition index with additional data \newcommand{\arrow}{ \begin{tikzpicture} \draw (3,0)--(3.07,.07); \draw (3,0)--(3.07,-.07); \end{tikzpicture} } %%% Arrow picture %%% % %\newcommand{\smallydcell}{ %\begin{tikzpicture} %\draw (0,0) rectangle (0.15,0.15); %\end{tikzpicture} %} %%%%%%%%%%% MACROS for spin Heisenberg category project %%%%%%%%%%%%%%%%% \newcommand{\ds}{\displaystyle} %%% Shortcut for display style \newcommand{\Heis}{\mathcal{H}_{tw}} %%% Twisted Heisenberg category \newcommand{\Tr}{\operatorname{Tr}} %%% Trace operator \newcommand{\TrH}{\Tr(\Heis)} %%% Trace of twisted Heisenberg category \newcommand{\EndPm}{End_{\Heis}(P^m)} %%% Endomorphism algebra for all upward strands \newcommand{\EndQn}{End_{\Heis}(Q^n)} %%% Endomorphism algebra for all downward strands \newcommand{\Endid}{\text{End}_{\Heis}(\UnitModule)} %%% Center of twisted Heisenberg category \newcommand{\Ser}{\mathbb{S}_n} %%% Sergeev algebra \newcommand{\Clif}{\mathcal{C\ell}_n} %%% Clifford algebra with n generators \newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}} %%% Add counter to equation \newcommand{\gr}{\operatorname{gr}} %%% Associated graded operator \newcommand{\bideg}{\operatorname{bideg}} %%% bi-degree symbol \newcommand{\grD}{\gr (\mathcal{D}^-)} %%% Associated graded D^- \newcommand{\TrHEv}{\TrH_{\overline{0}}} %%% Even part of trace \newcommand{\TrHOm}{\Tr^\omega(\mathcal{H}^{tw})_{\overline{0}}} \newcommand{\TrHPl}{\Tr^>(\mathcal{H}^{tw})_{\overline{0}}} \newcommand{\TrHMi}{\Tr^<(\mathcal{H}^{tw})_{\overline{0}}} \newcommand{\Ind}{\operatorname{Ind}} %%% Second induction operator \newcommand{\regrep}{\tau} %%% Character for regular representation \newcommand{\Sern}[1]{\MB{S}_{#1}} %%% Sergeev algebra of specified size \newcommand{\supersym}{\Gamma} %%% Supersymmetric functions \newcommand{\powersum}[1]{\tilde{p}^\#_{#1}} %%% Power supersymmetric functions \newcommand{\partsofsizen}[2]{m_{#2}(#1)} %%% Parts of size #2 in partition #1. \newcommand{\oddpartitions}[1][]{\mathcal{OP}_{#1}} %%% Odd partitions of size #1 \newcommand{\strictpartitions}[1][]{\mathcal{SP}_{#1}} %%% Strict partitions of size #1 \newcommand{\strictpartitionsodd}[1][]{\mathcal{SP}_{#1}^-} %%% Strict partitions with length odd \newcommand{\stanshiftedstrict}[1]{g_{#1}} %%% # of standard shifted strict Young diagrams of shape #1 \newcommand{\Wminus}{W^{-}} %%% Subalgebra of W^- \newcommand{\Winfty}{W_{1+ \infty}} %%% Vertex algebra W_{1+\infty} \newcommand{\length}[1]{\ell(#1)} %%% length of a partition \newcommand{\Schurgraph}{\mathbb{G}} %%% Schur's graph \newcommand{\pathsinSchur}[2]{h(#1,#2)} %%% Paths in Schur graph from #1 to #2 \newcommand{\pathsfromzero}[1]{h(#1)} %%% Paths in Schur graph from \emptyset to #1 \newcommand{\shifted}[1]{S(#1)} %%% Shifted Young diagram \newcommand{\lengthparity}{\delta} %%% Parity indicator for length of a partition \newcommand{\downtransition}[2]{p^{\downarrow}(#1,#2)} %%% Down transition function \newcommand{\uptransition}[2]{p^{\uparrow}(#1,#2)} %%% Up transition function \newcommand{\edgemulti}[2]{\kappa(#1,#2)} %%% The number of edges in Schur's graph between #1 and #2 \newcommand{\Plancherel}[1]{Pl_{#1}} %%% Plancherel measure on level #1 \newcommand{\SimpleSer}[1]{L^{#1}} %%% Simple Sergeev module corresponding to strict partition #1. \newcommand{\shiftedpowersum}{\mathfrak{p}} \newcommand{\KerovCoorUp}[1]{X(#1)} %%% The "up" Kerov interlacing coordinates \newcommand{\KerovCoorDown}[1]{Y(#1)} %%% The "down" Kerov interlacing coordinates \newcommand{\KerovCoorUpNoZero}[1]{X'(#1)} %%% The "up" Kerov coordinates without zero \newcommand{\JMeig}[1]{s(#1)} %%% Eigenvalue for JM element in Clifford-Hecke. \newcommand{\upmoment}[1]{\mathbf{g}^{\uparrow}_{#1}} %%% "Moments" for up transition measure. \newcommand{\downmoment}[1]{\mathbf{g}^{\downarrow}_{#1}} %%% "Moments" for down transition measure. \newcommand{\typeBWeyl}[1]{B_{#1}} %%% Hyperoctahedral group \newcommand{\hyperoctahedral}[1]{\widehat{B}_{#1}} %%% twisted Hyperoctahedral group \newcommand{\hyperoctohedral}[1]{\widehat{B}_{#1}} %%% twisted Hyperoctahedral group \newcommand{\cliffordgroup}[1]{\Pi_{#1}} %%% "Clifford group" \newcommand{\evencenter}[1]{Z(\Sern{#1})_{\overline{0}}} %%% Even center of Sergeev \newcommand{\distinguishedperm}[2]{\sigma_{#1;#2}} %%% Distinguished permutation \newcommand{\orbit}[1]{\mathrm{Conj}(#1)} %%% Orbit of distinguished perm under action of hyperoctohedral \newcommand{\octoclasssum}[2]{\wh{A}_{#1;#2}} %%% Conjugacy class sum in twisted octohedral group \newcommand{\Sergeevclasssum}[2]{A_{#1;#2}} %%% Sergeev class sum \newcommand{\centralidem}[1]{e_{#1}} %%% Central idempotent for Sergeev algebra \newcommand{\Heisgen}[1]{h_{#1}} %%% Heisenberg generator of trace \newcommand{\Virasorogen}[1]{h^{(x_1 + x_2)}_{#1}} %%% The Virasoro generator of the trace \newcommand{\PSchurFact}[1]{P^*_{#1}} %%% The factorial P-Schur function. \newcommand{\THeisencat}{\MC{H}^{t}'} %%% Twisted Heisenberg category (idempotent version) \newcommand{\dbar}[1]{\bar{d}_{#1}} %%% Counterclockwise bubbles \newcommand{\symtoHom}[1]{\MC{T}_{#1}} %%% Homomorphism from Sym to Hom(P^n) \newcommand{\FockSpaceCat}[1][]{\mathfrak{S}_{#1}} %%% Fock space category from Sergeev \newcommand{\FockSpaceFunctor}[1]{F^{\Heis}_{#1}} %%% Fock space functors of index #1 \newcommand{\hyperSergeevProj}[1]{\pi_{#1}} %%% Projection from hyperoctohedral to Sergeev \newcommand{\LcosHyper}[2]{\wh{\MC{LC}}^{#1}_{#2}} %%% Left coset representatives of hyperocto{n} in hyperocto{k} \newcommand{\LcosSer}[2]{\MC{LC}^{#1}_{#2}} %%% Left coset representatives of Sergeev(k) in Sergeev(n) \newcommand{\HyperChar}[1]{\wh{\chi}^{#1}} %%% Character of hyperocto algebra \newcommand{\SerChar}[1]{\chi^{#1}} %%% Character of Sergeev algebra \newcommand{\SimpleHyper}[1]{\wh{L}^{\lambda}} %%% Simple representation of hyperoctohedral group \newcommand{\normSerChar}[1]{\widetilde{\chi}^{#1}} %%% Normalized character for Sergeev \newcommand{\primaryisom}{\p} %%% Primary isomorphism of the paper. \newcommand{\funonYD}{\text{Fun}(\strictpartitions,\MB{C})} %%% Functions on partitions \newcommand{\supp}{\text{supp}} %%% Support of a permutation \newcommand{\distinguishedhyper}[2]{\omega^{#1}_{#2}} %%% Distinguished coset representative in the hyperoctahedral \newcommand{\supersymhom}[1]{F^{\supersym}_{#1}} %%% Fock space functors of index #1 \newcommand{\nonstrictodd}{D_{ns}} \newcommand{\oddindices}{D_{so}} \newcommand{\strictindices}{D_{ss}} \newcommand{\antiauto}{\varphi} %%% Antiautomorphism for Sergeev algebras \newcommand{\refequal}[1]{\xy {\ar@{=}^{#1} (-1,0)*{};(1,0)*{}}; \endxy} % shortcut to make arrow lines \newcommand{\arrowlines}{% \begin{tikzpicture} \draw[thick] (0,0) -- (-.08,-.08); \draw[thick] (0,0) -- (.08,-.08); \end{tikzpicture}% } \newcommand{\redarrowlines}{% \begin{tikzpicture} \draw[thick,red] (0,0) -- (-.08,-.08); \draw[thick,red] (0,0) -- (.08,-.08); \end{tikzpicture}% } % shortcut to make a Young cell for sub/superscripts \newcommand{\smallydcell}{ \begin{tikzpicture} \draw (0,0) rectangle (.2,.2); \end{tikzpicture} } %picture of trace of identity on P \newcommand{\trup}{ \begin{tikzpicture}[baseline=(current bounding box).center] \draw[thick] (2,2) circle (.5cm); \node at (1.5,2) {\arrowlines}; \node at (2,2) {$*$}; \end{tikzpicture}} %picture of trace of identity on Q \newcommand{\trdown}{ \begin{tikzpicture}[baseline=(current bounding box).center] \draw[thick] (2,2) circle (.5cm); \node[rotate = 180] at (1.5,2) {\arrowlines}; \node at (2,2) {$*$}; \end{tikzpicture}} %picture of \alpha_{\rho} in a box \newcommand{\tralpharho}{ \begin{tikzpicture}[baseline=(current bounding box).center] \draw[thick] (0,0) rectangle (.5,.5); \node at (.25,.25) {$\alpha_{\rho}$}; \end{tikzpicture}} %picture of clockwise bubble with k dots \newcommand{\ckpicture}{ \begin{tikzpicture} \draw[thick] (2,2) circle (.5cm); \node at (1.5,2) {\arrowlines}; \draw[fill=black] (1.62,2.33) circle (.08cm); \node at (1.3,2.7) {$2k$}; \end{tikzpicture}} %picture of counterclockwise bubble \newcommand{\cktildepicture}{ \begin{tikzpicture} \draw[thick] (2,2) circle (.5cm); \node[rotate = 180] at (1.5,2) {\arrowlines}; \draw[fill=black] (1.62,2.33) circle (.08cm); \node at (1.3,2.7) {$2k$}; \end{tikzpicture}} %\omega_{(1,0)}/\sqrt{2} \newcommand{\op}{\omega_{+}} %\omega_{(-1,0)}/\sqrt{2} \newcommand{\om}{\omega_{-}}