Difference between revisions of "06-1350/Class Notes for Thursday November 16"

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* <math>{\mathcal A}(\uparrow_n)</math> is a VS-algebra (see more at [[VS, TS and TG Algebras]]).
 
* <math>{\mathcal A}(\uparrow_n)</math> is a VS-algebra (see more at [[VS, TS and TG Algebras]]).
 
* In the coordinates above, write the <math>TR\Phi B</math> relations in various algebraic notations.
 
* In the coordinates above, write the <math>TR\Phi B</math> relations in various algebraic notations.
** R4: <math>(1230)^\star B^\pm\cdot(1213)^\star B^\pm\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B^\pm</math> or <math>(B^\pm_{1a}B^\pm_{2a}\Phi_{1a}; B^\pm_{1b}B^\pm_{2b}; B^\pm_{1c}B^\pm_{2b}\Phi_{1b}; B^\pm_{2c}\Phi_{1c}) = (\Phi_{2a}B^\pm_{3a}; \Phi_{2a}B^\pm_{3b}; \Phi_{2b}B^\pm_{3c}; \Phi_{2c}B^\pm_{3c})</math>.
+
** R4: <math>(1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B</math> or <math>(B_{1a}B_{2a}\Phi_{1a}; B_{1b}B_{2b}; B_{1c}B_{2a}\Phi_{1b}; B_{2c}\Phi_{1c}) = (\Phi_{2a}B_{3a}; \Phi_{2a}B_{3b}; \Phi_{2b}B_{3c}; \Phi_{2c}B_{3c})</math>.
 +
** R3: <math>(1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star B = (1123)^\star B\cdot(1203)^\star B\cdot(1231)^\star B</math> or <math>(B_{1a}B_{2a}B_{3a}; B_{1b}B_{2b}; B_{1c}B_{2a}B_{3b}; B_{2c}B_{3c}) = (B_{4a}B_{5a}B_{6a}; B_{4a}B_{5b}B_{6b}; B_{4b}B_{6c}; B_{4c}B_{5c}B_{6a})</math>.
 +
** R2, R1
 
** <math>B^{\pm}</math> in terms of <math>\Phi</math> and <math>R</math> and <math>R</math> in terms of <math>T</math>.
 
** <math>B^{\pm}</math> in terms of <math>\Phi</math> and <math>R</math> and <math>R</math> in terms of <math>T</math>.
** R3, R2, R1
 
 
** Symmetry of <math>\Phi</math> and of <math>B^{\pm}</math>.
 
** Symmetry of <math>\Phi</math> and of <math>B^{\pm}</math>.
 
** <math>u</math>, <math>d</math> and <math>\#</math>
 
** <math>u</math>, <math>d</math> and <math>\#</math>
 
** Idempotence for <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>.
 
** Idempotence for <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>.

Revision as of 22:36, 15 November 2006

In Preparation

The information below is preliminary and cannot be trusted! (v)

Today's Agenda

  • Sweeping clean a tree and {\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)}).
06-1350-TRPhiB.png
  • {\mathcal A}(\uparrow_n) is a VS-algebra (see more at VS, TS and TG Algebras).
  • In the coordinates above, write the TR\Phi B relations in various algebraic notations.
    • R4: (1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B or (B_{1a}B_{2a}\Phi_{1a}; B_{1b}B_{2b}; B_{1c}B_{2a}\Phi_{1b}; B_{2c}\Phi_{1c}) = (\Phi_{2a}B_{3a}; \Phi_{2a}B_{3b}; \Phi_{2b}B_{3c}; \Phi_{2c}B_{3c}).
    • R3: (1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star B = (1123)^\star B\cdot(1203)^\star B\cdot(1231)^\star B or (B_{1a}B_{2a}B_{3a}; B_{1b}B_{2b}; B_{1c}B_{2a}B_{3b}; B_{2c}B_{3c}) = (B_{4a}B_{5a}B_{6a}; B_{4a}B_{5b}B_{6b}; B_{4b}B_{6c}; B_{4c}B_{5c}B_{6a}).
    • R2, R1
    • B^{\pm} in terms of \Phi and R and R in terms of T.
    • Symmetry of \Phi and of B^{\pm}.
    • u, d and \#
    • Idempotence for T, R, \Phi and B^{\pm}.