Frobenius image
The Frobenius characteristic of a character $\chi$ of the symmetric group $S_n$ is $$\mathcal F(\chi) = \frac{1}{n!}\sum_{\sigma\in S_n} \chi(\sigma) p_{ct(\sigma)},$$ where $ct(\sigma)$ is the cycle type of $\sigma$ and $p_\lambda$ is the power sum symmetric function indexed by the partition $\lambda$.
{{{id=48| /// }}} {{{id=46| /// }}} {{{id=43| /// }}} {{{id=45| /// }}} {{{id=55| /// }}} {{{id=54| /// }}} {{{id=53| /// }}} {{{id=52| /// }}} {{{id=51| /// }}} {{{id=50| /// }}} {{{id=49| /// }}} {{{id=69| /// }}} {{{id=68| /// }}} {{{id=67| /// }}} {{{id=66| /// }}} {{{id=65| /// }}}The Frobenius characteristic of a character $\chi$ of the symmetric group $S_n$ is $$\mathcal F(\chi) = \frac{1}{n!}\sum_{\lambda \vdash n} \frac{\chi(\sigma)}{z_\lambda} p_{\lambda},$$ where $\sigma$ is a permutation of cycle type $\lambda$ and for $\lambda = 1^{a_1} 2^{a_2} \ldots$, $$z_\lambda =1^{a_1}a_1!2^{a_2}a_2!\ldots.$$
{{{id=44| def symmetric_group_conjugacy_class_representative(partition): r""" Return a representative of the conjugacy class of the symmetric group index by partition. """ n = sum(partition) rep = range(2,n+2) sum1 = -1 sum2 = 1 for pi in partition: sum1 += pi rep[sum1] = sum2 sum2 += pi return Permutation(rep) /// }}} {{{id=40| def frobenius_characteristic_2(chi, n, ring=QQ): r""" The Frobenius image of a character of the symmetric group. INPUT: - chi -- callable, takes a permutation and returns the value of the character on the permutation - n -- integer, the degree of the underlying symmetric group - ring -- ring, the ring over which to define the symmetric functions. EXAMPLES:: sage: n = 4 sage: s = SFASchur(QQ) sage: for rho in SymmetricGroupRepresentations(n): ... print s(frobenius_image_character(rho.to_character(), n=n)) ... s[4] s[3, 1] s[2, 2] s[2, 1, 1] s[1, 1, 1, 1] """ p = SFAPower(ring) Sym = Permutations(n) frob = p.zero() for partition in Partitions(n): sigma = symmetric_group_conjugacy_class_representative(partition) zee = prod([(i+1)**ai*factorial(ai) for (i,ai) in enumerate(partition.to_exp())]) frob += QQ(chi(sigma))/zee * p(partition) return frob /// }}} {{{id=58| n = 4 s = SFASchur(QQ) for rho in SymmetricGroupRepresentations(n): print s(frobenius_characteristic_2(rho.to_character(), n=n)) /// }}} {{{id=57| /// }}} {{{id=56| /// }}} {{{id=83| /// }}} {{{id=82| /// }}} {{{id=81| /// }}} {{{id=80| /// }}} {{{id=79| /// }}} {{{id=78| /// }}} {{{id=77| /// }}} {{{id=76| /// }}} {{{id=75| /// }}} {{{id=74| /// }}} {{{id=73| /// }}} {{{id=72| /// }}}
If $e \in \mathbb QS_n$ is an idempotent, then the Frobenius characteristic of the $S_n$-module $$\mathbb QS_n e$$ is given by $$\sum_{\sigma\in S_n} coeff(\sigma, e) \, p_{ct(\sigma)},$$ where $coeff(\sigma,e)$ is the coefficient of $\sigma$ in $e$.
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