# Defining several independent Weyl Character Rings

I have the following problem. I want to define expression which contains elements of several Weyl Character Rings. The problem comes from physics. I have expression for theory with several symmetries, which can be of the same or different groups which involves characters of these groups representations. Let's take a simplest example of such expression

$x = \chi_{R_1}(U)\chi_{R_2}(V)$

where $U\in G_1$, $V\in G_2$ are matrices in groups $G_1$ and $G_2$ and $R_{1,2}$ are corresponding representations. Let's say I want to find $x^2$ (or any other power) expanded in terms of irreps. If $G_1 \neq G_2$ I can write the following code that will work

G1 = WeylCharacterRing(['A',2], base_ring=QQ, style="coroots")
G2 = WeylCharacterRing(['A',1], base_ring=G1, style="coroots")
fund1 = G1(G1.fundamental_weights()[1]);
fund2 = G2(G2.fundamental_weights()[1]);
x  = fund1*fund2
x**2


which gives me what I want

  (A2(0,1)+A2(2,0))*A1(0) + (A2(0,1)+A2(2,0))*A1(2)


Here for simplicity I took $SU(2)$ and $SU(3)$ groups and fundamental representations for both of them. Now if $G_1 = G_2$ (for example $A_2$ root system in both cases) it also works but result is not very readable now since I can not understand which representation belongs to one group or another. For example

G1 = WeylCharacterRing(['A',2], base_ring=QQ, style="coroots")
G2 = WeylCharacterRing(['A',2], base_ring=G1, style="coroots")
fund1 = G1(G1.fundamental_weights()[1]);
fund2 = G2(G2.fundamental_weights()[1]);
x**2


gives me back

(A2(0,1)+A2(2,0))*A2(0,0) + (A2(0,1)+A2(2,0))*A2(0,3) + (2*A2(0,1)+2*A2(2,0))*A2(1,1) + (A2(0,1)+A2(2,0))*A2(3,0) + (A2(0,1)+A2(2,0))*A2(2,2)


which is hard to read. Is there a simple way to keep track which of the expressions belongs to one group or another?

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It's hard to read but not impossible: in each term the first factor in parentheses is in the base ring G1, and the second factor is in G2.

( 2024-08-09 09:26:59 +0200 )edit

@rburing Yes sure it is possible to read. The problem that real expressions I work with can have more groups and will be more massive in general. It is still possible to read them since order of factors is always the same but I was wondering if there is a smart way to simplify readability.

( 2024-08-09 13:51:59 +0200 )edit

@FrédéricC Thanks a lot, that is exactly what I was looking for.

( 2024-08-09 23:41:45 +0200 )edit

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you can use the prefix as follows

sage: A=WeylCharacterRing(['A',2], base_ring=QQ, style="coroots", prefix="🍌")

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