example 5 from Murai-Raicu

BettiCharacters.m2

@@ -1278,6 +1278,7 @@ Node
       	"BettiCharacters Example 2"
       	"BettiCharacters Example 3"
       	"BettiCharacters Example 4"
+	"BettiCharacters Example 5"
 
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     Key
@@ -1608,6 +1609,63 @@ Node
        	    character(B,{1,2})
 
 
+Node
+    Key
+    	"BettiCharacters Example 5"
+    Headline
+    	semidirect product of torus and symmetric group
+    Description
+    	Text
+	    We present the example in the introduction of
+	    @HREF("https://doi.org/10.1112/jlms.12551",
+	    "S. Murai, C. Raicu - An equivariant Hochster’s formula for $\\mathfrak{S}_n$-invariant monomial ideals")@.
+
+	    Consider the ideal $I$ in three variables generated by
+	    monomials whose exponent vectors are permutations of
+	    $(4,1,1)$ or $(5,2,0)$. This ideal is clearly stable
+	    under the permutation action of $\mathfrak{S}_3$.
+	    Moreover, $I$ is compatible with the fine grading
+	    on $R = \Bbbk [x_1,x_2,x_3]$ given by $\deg (x_i) = e_i
+	    \in \mathbb{Z}^3$. We compute a minimal free resolution of
+	    $R/I$ and show its Betti diagram.
+    	Example
+    	    R = QQ[x_1..x_3,Degrees=>{{1,0,0},{0,1,0},{0,0,1}}]
+	    I = ideal(x_1^4*x_2*x_3,x_1*x_2^4*x_3,x_1*x_2*x_3^4,
+		x_1^5*x_2^2,x_1^5*x_3^2,x_1^2*x_2^5,x_1^2*x_3^5,x_2^5*x_3^2,x_2^2*x_3^5)
+	    RI = res I
+	    betti RI
+	Text
+	    Next, we set up the action of the semidirect product
+	    $(\Bbbk^\times)^3 \rtimes \mathfrak{S}_3$ where
+	    $\mathfrak{S}_3$ acts on $(\Bbbk^\times)^3$ by
+	    permuting entries. This results in $\mathfrak{S}_3$
+	    acting on the grading group $\mathbb{Z}^3$ (the character
+		group of $(\Bbbk^\times)^3$) by permuting the
+	    entries of the degree vectors. Thus, the orbit of a
+	    degree $d\in \mathbb{Z}^3$ consists of all permutations
+	    of $d$; we fix the nonincreasing permutation of $d$ as
+	    the distinguished representative of this orbit.
+	    See @TO "Semidirect"@ for details.
+	Example
+	    S3 = symmetricGroupActors(R)
+    	    A = action(RI,S3,Semidirect=>{uniquePermutations,rsort})
+    	    c = character A
+    	Text
+    	    To match the description of the paper, which resolves the
+	    ideal $I$ instead of the quotient $R/I$, we remove
+	    the component in homological degree 0, then shift the
+	    complex to the left. Finally, the resulting character is
+	    decomposed against the character table of $\mathfrak{S}_3$.
+    	Example
+	    c = (c - character(R,3,hashTable{(0,{0,0,0})=>matrix{{1,1,1}}}))[1]
+    	    T = symmetricGroupTable(3,QQ)
+	    decomposeCharacter(c,T)
+	Text
+	    The irreducible representations found above match
+	    our expectations as can be verified by
+	    applying Pieri's rule to the description
+	    in Example 1.4 of Murai and Raicu's paper.
+
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