The symmetric functions catalog

#Specht modules

#Introduction to Specht modules

The Specht module \(S^\lambda\) is the basic irreducible \(\symS_n\)-module attached to a partition \(\lambda\vdash n.\) Over \(\setC,\) the modules \(S^\lambda,\) as \(\lambda\) ranges over all partitions of \(n,\) form a complete set of pairwise non-isomorphic irreducible \(\symS_n\)-modules. Their characters are sent by the Frobenius characteristic to the Schur functions: \[\frobChar(S^\lambda)=\schurS_\lambda.\] The construction below is the standard tableau construction; see [Ful97, Sag01, JK84].

#Tabloids

In order to construct an \(\symS_n\)-module, we first need a vector space on which \(\symS_n\) acts. A tableau of shape \(\lambda\) is a filling of the Young diagram \(\lambda\) with \(\{1,2,3,\dotsc,n\},\) using each entry once. The row-stabilizer \(R_T\) of a tableau \(T\) is the set of permutations that keep entries within rows. Similarly, the column-stabilizer \(C_T\) of \(T\) is the set of permutations that keep entries within columns.

For example, the following tableaux are in the same orbit under \(R_T,\) and there are \(3! \cdot 2!\) permutations total in \(R_T.\)

Similarly, the tableaux which can be obtained from the first one by some element of \(C_T\) are the following.

Two tableaux are considered equivalent if they are in the same orbit under the row-stabilizer. An equivalence class is called a tabloid and is denoted by \(\{T\}.\) Equivalently, a tabloid remembers the set of entries in each row, but forgets the order inside each row.

The permutation module \(M^\lambda\) is the \(\setC\)-vector space with basis all tabloids of shape \(\lambda.\) The symmetric group acts by permuting the entries: \[\sigma \{T\} = \{\sigma T\}.\] This action is well-defined because row-equivalent tableaux remain row-equivalent after applying \(\sigma.\)

#Polytabloids

A polytabloid is obtained by antisymmetrizing a tabloid in its columns. For a tableau \(T,\) define \[e_T \coloneqq \sum_{\tau\in C_T}\sign(\tau)\{\tau T\}.\] The Specht module of shape \(\lambda\) is \[S^\lambda \coloneqq \langle e_T : T\text{ is a tableau of shape }\lambda\rangle_{\setC} \subseteq M^\lambda.\] Since \(\sigma e_T=e_{\sigma T},\) the space \(S^\lambda\) is stable under the action of \(\symS_n.\)

#Frobenius characteristic

The Specht modules explain why Schur functions are the natural character basis for \(\symS_n\)-representations. If \(\chi^\lambda\) is the irreducible character of \(S^\lambda,\) then \[\frobChar(\chi^\lambda)=\schurS_\lambda.\] Consequently, if an \(\symS_n\)-module decomposes as \[M \cong \bigoplus_{\lambda\vdash n} (S^\lambda)^{\oplus c_\lambda},\] then its Frobenius characteristic is \[\frobChar(M)=\sum_{\lambda\vdash n} c_\lambda\schurS_\lambda.\] Thus Schur positivity is the statement that a symmetric function is the Frobenius characteristic of an honest \(\symS_n\)-module, or at least behaves as if it were one.

The dimension of \(S^\lambda\) is the number \(f^\lambda\) of standard Young tableaux of shape \(\lambda,\) given by the hook formula.

#Higher Specht polynomials

Specht polynomials realize the same irreducible modules inside a polynomial ring. If \(T\) is a tableau, its Specht polynomial is \[g_T \coloneqq \prod_{C\text{ a column of }T} \prod_{\substack{i\text{ above }j\\i,j\in C}}(x_i-x_j).\] As \(T\) ranges over the standard tableaux of shape \(\lambda,\) these polynomials span a copy of \(S^\lambda.\)

Higher Specht polynomials replace the row-number exponents behind \(g_T\) by a more general filling. Let \(A\) be a filling of \(\lambda\) by nonnegative integers and define \[x_T^A\coloneqq\prod_{b\in\lambda}x_{T(b)}^{A(b)}, \qquad \varepsilon_T \coloneqq \sum_{\tau\in C_T}\sum_{\sigma\in R_T} \sign(\tau)\tau\sigma.\] The corresponding higher Specht polynomial is \[\defin{F_T^A}\coloneqq\varepsilon_T(x_T^A).\] A higher Specht basis of an \(\symS_n\)-module is a basis partitioned into families indexed by partitions, with each family transforming like the standard Specht-polynomial basis of the corresponding irreducible module.

For a standard tableau \(S,\) let \(\operatorname{cc}(S)\) be its cellwise cocharge filling and abbreviate \(F_T^S\coloneqq F_T^{\operatorname{cc}(S)}.\) These polynomials give the classical higher Specht basis of the coinvariant ring and also provide candidates for many generalized coinvariant rings.

The Garsia–Procesi module attached to \(\mu\vdash n\) is the quotient \[R_\mu\coloneqq\setC[x_1,\dotsc,x_n]/I_\mu,\] where \(I_\mu\) is the Tanisaki ideal. It is isomorphic to the cohomology of the Springer fiber of Jordan type \(\mu,\) with its Springer \(\symS_n\)-action. Write \(\operatorname{ctype}(S)\) for the partition obtained from the catabolizability algorithm.

The same paper gives useful nonvanishing tests before passing to a quotient. For fixed \(A,\) either all \(F_T^A\) are zero or all are nonzero; in the latter case they span a copy of \(S^\lambda.\) A semistandard exponent filling \(A\) always gives nonzero polynomials. When \(\lambda\) has two columns, \(F_T^A\neq0\) precisely when entries of \(A\) can be permuted within rows so that neither column contains a repeated entry [CGH26, Prop. 1.6, Cor. 1.14, Prop. 3.26].

The shape restriction in the basis theorem reflects a real change in the known constructions. The Gillespie–Rhoades candidate can contain the zero polynomial for the five-row shape \((2,2,2,2,2),\) and the two candidate bases already differ for the four-row shape \((2,2,2,2).\) The Chou–Hanada basis remains conjectural for general shapes.

#Comparison with the 0-Hecke picture

The 0-Hecke algebra gives a useful parallel to the classical Specht-module story. The group algebra \(\setC[\symS_n]\) has irreducible modules indexed by partitions, and its Frobenius characteristic lands in symmetric functions. The 0-Hecke algebra \(H_n(0)\) has simple modules indexed by compositions, and its quasisymmetric characteristic lands in quasisymmetric functions.

\( \text{Feature} \) \( \setC[\symS_n] \) \( H_n(0)\) \( \text{Simple modules} \) \( S^\lambda,\ \lambda\vdash n \) \( F^\alpha,\ \alpha\vDash n\) \( \text{Characteristic} \) \( \frobChar(S^\lambda)=\schurS_\lambda \) \( \mathrm{Ch}(F^\alpha)=\gessel_\alpha\) \( \text{Indexing objects} \) \( \text{partitions and standard tableaux} \) \( \text{compositions and descent sets}\) \( \text{Dual Hopf algebra} \) \( \spaceSym \) \( \mathrm{NSym}\)

This dictionary is one reason that the QSym–NSym duality is best understood together with ordinary representation theory of \(\symS_n.\)

#Related directions

For related work, R. Paget and M. Wildon construct homomorphisms from Specht modules into Foulkes modules [PW11]. R. Hodges and H. Yin give a non-iterative straightening algorithm for skew Schur modules, a closely related tableau-based construction [HY25].

J. Kim constructs a rotation-invariant web basis for the Specht modules \(S^{(d^3,1^{n-3d})},\) indexed by normal plabic graphs or augmented \(\mathrm{SL}_3\) webs [Kim24]. The construction gives skein relations for the \(\symS_n\)-action and leads to a cyclic-sieving result for the corresponding \(q\)-hook length formula.

M. Gillespie studies higher Specht polynomials in two sets of variables under the diagonal action of \(\symS_n\) [Gil24]. In particular, she constructs a higher Specht basis for Garsia–Haiman modules for hook shapes. In a complementary direction, S. Zemel generalizes higher Specht polynomials to homogeneous representations and quotients related to the rings \(R_{n,k,s}\) [Zem25].

Bibliography

1. [CGH26]Raymond Chou, Maria Gillespie and Mitsuki Hanada. Nonvanishing higher Specht polynomials and a construction for three row and hook shape Garsia–Procesi modules. arXiv:2609.12255v1, 2026.
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   @article{ChouGillespieHanada2026x,
     author = {Raymond Chou and Maria Gillespie and Mitsuki Hanada},
     title = {Nonvanishing higher {S}pecht polynomials and a construction for
       three row and hook shape {G}arsia--{P}rocesi modules},
     year = {2026},
     eprint = {2609.12255v1},
     url = {https://arxiv.org/abs/2609.12255v1},
     journal = {arXiv e-prints}
   }
   

2. [Ful97]William Fulton. Young tableaux: With applications to representation theory and geometry. London mathematical society student texts (book 35). Cambridge University Press, 1997.
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   @book{Fulton1997,
     title = {Young Tableaux: With Applications to Representation Theory and Geometry},
     doi = {10.1017/cbo9780511626241},
     author = {William Fulton},
     isbn = {978-0511626241},
     series = {London Mathematical Society Student Texts (Book 35)},
     year = {1997},
     publisher = {Cambridge University Press}
   }
   

3. [Gil24]Maria Gillespie. Higher Specht polynomials under the diagonal action. arXiv:2402.05221, 2024.
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   @article{Gillespie2024x,
     author = {Maria Gillespie},
     title = {Higher {S}pecht polynomials under the diagonal action},
     year = {2024},
     eprint = {2402.05221},
     url = {https://arxiv.org/abs/2402.05221},
     journal = {arXiv e-prints}
   }
   

4. [HY25]Reuven Hodges and Hanzhang Yin. A non-iterative straightening algorithm and orthogonality for skew Schur modules. arXiv:2511.03702, 2025.
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   @article{HodgesYin2025x,
     author = {Reuven Hodges and Hanzhang Yin},
     title = {A non-iterative straightening algorithm and orthogonality for skew {S}chur modules},
     year = {2025},
     eprint = {2511.03702},
     url = {https://arxiv.org/abs/2511.03702},
     journal = {arXiv e-prints}
   }
   

5. [JK84]Gordon James and Adalbert Kerber. The representation theory of the symmetric group. Cambridge University Press, 1984.
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   @book{JamesKerber1984,
     doi = {10.1017/cbo9781107340732},
     url2 = {https://doi.org/10.1017/cbo9781107340732},
     year = {1984},
     publisher = {Cambridge University Press},
     author = {Gordon James and Adalbert Kerber},
     title = {The Representation Theory of the Symmetric Group}
   }
   

6. [Kim24]Jesse Kim. Rotation invariant webs for three row flamingo Specht modules. arXiv:2402.11994, 2024.
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   @article{Kim2024x,
     author = {Jesse Kim},
     title = {Rotation invariant webs for three row flamingo {S}pecht modules},
     year = {2024},
     eprint = {2402.11994},
     url = {https://arxiv.org/abs/2402.11994},
     journal = {arXiv e-prints}
   }
   

7. [PW11]Rowena Paget and Mark Wildon. Set families and Foulkes modules. Journal of Algebraic Combinatorics, 34(3):525–544, 2011.
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   @article{PagetWildon2011,
     author = {Rowena Paget and Mark Wildon},
     title = {Set families and {F}oulkes modules},
     journal = {Journal of Algebraic Combinatorics},
     volume = {34},
     number = {3},
     pages = {525--544},
     year = {2011},
     doi = {10.1007/s10801-011-0282-8}
   }
   

8. [Sag01]Bruce E. Sagan. The symmetric group. Springer New York, 2001.
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   @book{Sagan2001,
     doi = {10.1007/978-1-4757-6804-6},
     url2 = {https://doi.org/10.1007/978-1-4757-6804-6},
     year = {2001},
     publisher = {Springer New York},
     author = {Bruce E. Sagan},
     title = {The Symmetric Group}
   }
   

9. [Zem25]Shaul Zemel. Generalized Higher Specht Polynomials and Homogeneous Representations of Symmetric Groups. arXiv:2505.07098, 2025.
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   @article{Zemel2025x,
     author = {Shaul Zemel},
     title = {Generalized {H}igher {S}pecht {P}olynomials and {H}omogeneous {R}epresentations of {S}ymmetric {G}roups},
     year = {2025},
     eprint = {2505.07098},
     url = {https://arxiv.org/abs/2505.07098},
     journal = {arXiv e-prints}
   }