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lr_tensor
Compute the tensor product of two Young tableaux
Compute the tensor product of two tableaux or filled tableaux. The
algorithm acts on objects which have the Tableau
or FilledTableau property, through which it is possible to
set the dimension. The standard Littlewoord-Richardson algorithm is
used to construct the tableaux in the tensor product. An example
with Tableau objects is given below.\tableau{#}::Tableau(dimension=10).
ex:=\tableau{2}{2} \tableau{2}{2}{1};
lr_tensor(_);
\(\displaystyle{}\ydiagram{2,2} \otimes \ydiagram{2,2,1}\)
\tableau(2 , 2) \tableau(2 , 2 , 1)
\(\displaystyle{}\ydiagram{4,4,1} \oplus \ydiagram{4,3,2} \oplus \ydiagram{4,3,1,1} \oplus \ydiagram{4,2,2,1} \oplus \ydiagram{3,3,2,1} \oplus \ydiagram{3,3,1,1,1} \oplus \ydiagram{3,2,2,2} \oplus \ydiagram{3,2,2,1,1} \oplus \ydiagram{2,2,2,2,1}\)
\tableau(4 , 4 , 1) + \tableau(4 , 3 , 2) + \tableau(4 , 3 , 1 , 1) + \tableau(4 , 2 , 2 , 1) + \tableau(3 , 3 , 2 , 1) + \tableau(3 , 3 , 1 , 1 , 1) + \tableau(3 , 2 , 2 , 2) + \tableau(3 , 2 , 2 , 1 , 1) + \tableau(2 , 2 , 2 , 2 , 1)
The same example, but now with
FilledTableau objects, is\ftableau{#}::FilledTableau(dimension=10).
ex:=\ftableau{0,0}{1,1} \ftableau{a,a}{b,b};
\(\displaystyle{}\ytableaushort{{0}{0},{1}{1}} \otimes \ytableaushort{{a}{a},{b}{b}}\)
\ftableau({0, 0} , {1, 1}) \ftableau({a, a} , {b, b})
lr_tensor(_);
\(\displaystyle{}\ytableaushort{{0}{0}{a}{a},{1}{1}{b}{b}} \oplus \ytableaushort{{0}{0}{a}{a},{1}{1}{b},{b}} \oplus \ytableaushort{{0}{0}{a}{a},{1}{1},{b}{b}} \oplus \ytableaushort{{0}{0}{a},{1}{1}{b},{a},{b}} \oplus \ytableaushort{{0}{0}{a},{1}{1},{a}{b},{b}} \oplus \ytableaushort{{0}{0},{1}{1},{a}{a},{b}{b}}\)
\ftableau({0, 0, a, a} , {1, 1, b, b}) + \ftableau({0, 0, a, a} , {1, 1, b} , b) + \ftableau({0, 0, a, a} , {1, 1} , {b, b}) + \ftableau({0, 0, a} , {1, 1, b} , a , b) + \ftableau({0, 0, a} , {1, 1} , {a, b} , b) + \ftableau({0, 0} , {1, 1} , {a, a} , {b, b})
ex:=\ftableau{1} \ftableau{2} \ftableau{3} \ftableau{4};
\(\displaystyle{}\ytableaushort{{1}} \otimes \ytableaushort{{2}} \otimes \ytableaushort{{3}} \otimes \ytableaushort{{4}}\)
\ftableau(1) \ftableau(2) \ftableau(3) \ftableau(4)
converge(ex):
lr_tensor(_)
distribute(_)
;
\(\displaystyle{}\ytableaushort{{1}{2}{3}{4}} \oplus \ytableaushort{{1}{2}{3},{4}} \oplus \ytableaushort{{1}{2}{4},{3}} \oplus \ytableaushort{{1}{2},{3}{4}} \oplus \ytableaushort{{1}{2},{3},{4}} \oplus \ytableaushort{{1}{3}{4},{2}} \oplus \ytableaushort{{1}{3},{2}{4}} \oplus \ytableaushort{{1}{3},{2},{4}} \oplus \ytableaushort{{1}{4},{2},{3}} \oplus \ytableaushort{{1},{2},{3},{4}}\)
\ftableau({1, 2, 3, 4}) + \ftableau({1, 2, 3} , 4) + \ftableau({1, 2, 4} , 3) + \ftableau({1, 2} , {3, 4}) + \ftableau({1, 2} , 3 , 4) + \ftableau({1, 3, 4} , 2) + \ftableau({1, 3} , {2, 4}) + \ftableau({1, 3} , 2 , 4) + \ftableau({1, 4} , 2 , 3) + \ftableau(1 , 2 , 3 , 4)