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Greg Warrington, Aug 4, 2020
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The following files are plain pythoon --- no sage required
==========================================================
ah.py		-	class implementing abacus histories

lc.py		- 	class implementing basic Laurent polynomials for coefficients of abacus histories

ah_paper.py	-	computations illustrated in AH paper

SageMath is required for the following
======================================
pleth_basics.py	     -	 just making sure I understand basic of plethysm and how it's implemented

Calpha_plethysm.py   - 	 plethystic definitions of various creation operators
		     	 {S,C,B,H}_b - defined using e_c, h_c and their skews		     
			 {S,H,C,B}_b_pleth  - plethystic definitions of these operators
			 S_b_viaC, C_b_viaS, B_b_omega - alternate definitions for these operators

check_ah.py          -   code for checking various formulas for operators agree

=====================================================================================================

To compute, say, the result of applying the creation operator C_2 to the Schur function s_221:

    ab = AH.snu_abacus([3,1])
    ab.Cm_op(-2,LC([ab]),verbose=False).print_lc()

    The "LC([ab])" just converts the abacus history ab into an
    internal representation of a linear combination of abacus
    histories, each of whose coefficients is \pm a power of q.

The output in this case is:

X . X . . X 
|   |     | 
* X * . . * 
Coeff:  -q^3
            
X *-X . . X 
| |       | 
* * X . . * 
Coeff:  +q^2
            
X *-X . *-X 
| |     |   
* * . X * . 
Coeff:  +q^1
            
X *-X *-*-X 
| |   |     
* * . * X . 
Coeff:  -q^0
    
Reading off the partitions encoded by the bottom row of each abacus
history, we find that the answer is
-q^3 s_20 + q^2 s_20 + q s_11 - s_11

If we are interested only in the final expansion and not the abacus
histories themselves, we can instead run:

    ab = AH.snu_abacus([3,1])
    ab.Cm_op(-2,LC([ab]),verbose=False).print_lc_simplified()

which yields
(-1 + q^1) s_11 + (+ q^2 - q^3) s_2 

The function _convert_lc() in check_ah.py can be used to convert the
internal representation of such a linear combination abacus histories
into a proper expansion in terms of Schur functions whose coefficients
are Laurent polynomials in q.

To comput the result of applying a creation operator that is indexed
by a composition to 1, we run:

AH.H_alpha([3,1,2]).print_lc_simplified()

and get:

 + q^1 s_321  + q^2 s_33  + q^2 s_411 + (+ q^2 + q^3) s_42 + (+ q^3 + q^4) s_51  + q^5 s_6 

Analogous functions are defined for C_alpha and S_alpha.