In this example, we identify the Betti characters of the Specht ideal associated with the partition (5,2). The action of the symmetric group on the resolution of this ideal is described in K. Shibata, K. Yanagawa - Minimal free resolutions of the Specht ideals of shapes (n−2,2) and (d,d,1). The same ideal is also the ideal of the 6-equals subspace arrangement in a 7-dimensional affine space. This point of view is explored in C. Berkesch, S. Griffeth, S. Sam - Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k+1)-equals ideal where the action of the symmetric group on the resolution is also described.
We begin by constructing the ideal explicitly. As an alternative, the ideal can be obtained using the function spechtPolynomials provided by the package SpechtModule. We compute a minimal free resolution and its Betti table.
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Next we set up the group action on the resolution. The group is the symmetric group on 7 elements. Its conjugacy classes are determined by cycle types, which are in bijection with partitions of 7. Representatives for the conjugacy classes of the symmetric group acting on a polynomial ring by permuting the variables can be obtained via symmetricGroupActors. Once the action is set up, we compute the Betti characters.
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To make sense of these characters we decompose them against the character table of the symmetric group, which can be computed using the function symmetricGroupTable. The irreducible characters are indexed by the partitions of 7, which are written using a compact notation (the exponents indicate how many times a part is repeated).
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As expected from the general theory, we find a single irreducible representation in each homological degree.
Finally, we can observe the Gorenstein duality of the resolution and its character. We construct the character of the sign representation concentrated in homological degree 0, internal degree 7. Then we dualize the character of the resolution previously computed, shift its homological degree by the length of the resolution, and twist it by the sign character just constructed: the result is the same as the character of the resolution.
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The second argument in the dual command is the restriction of complex conjugation to the field of definition of the characters. For more information, see dual(Character,RingMap).