\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_{-}}
