Representation theory

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#Representation theory

#Representation of a group

Let $G$ be a group and $V$ a vector space. A representation is a map $\rho$ that sends elements in $G$ to linear maps on $V,$ with the condition that it is compatible with the group structure. To be precise, let $\psi_g : V \to V$ be the linear map on $V$ we associate with $g \in G.$ Then \[\psi_{g_1 g_2} = \psi_{g_1} \circ \psi_{g_2} \text{ for all } g_1,g_2 \in G.\] Thus, a representation is a group homomorphism $\rho : G \to \GL(V).$

Example

Let $G = \symS_n$ and $V=\setC[x_1,\dotsc,x_n].$ Note that $V$ is not finite-dimensional. Then we let $G$ act on $V$ by permuting the indices of the variables. After choosing a monomial basis of $V,$ the map $\rho$ sends a permutation $g \in G$ to a permutation matrix in $\GL(V).$

#Irreducible representation

Let $W \subset V$ be a subspace and let $\rho:G \to \GL(V)$ be a representation. If $\rho(g)w \in W$ for all $w \in W,$ then we say that $W$ is $G$-invariant.

If the only $G$-invariant subspaces of $V$ are $V$ itself and $\{0\},$ then we say that $\rho$ is an irreducible representation.

For a finite group $G$ over $\setC,$ every finite-dimensional representation can be decomposed as a direct sum of irreducible representations. That is, if $\rho: G \to \GL(V)$ is such a representation, we can write $V = V_1 \oplus \dotsb \oplus V_k$ such that for every $i$ we have \[\rho(g) v \in V_i \; \text{ for all }\; v \in V_i\] and the restricted representation on $V_i$ is irreducible.

#Tensor products

If $\rho_1$ and $\rho_2$ are two representations of $G,$ sending group elements to elements in $\GL(V)$ and $\GL(W),$ then we define the tensor product of representations as the map sending $g$ to the tensor product of the individual images: \[(\rho_1 \otimes \rho_2)(g) \coloneqq (\rho_1(g) \otimes \rho_2(g))\]

#Character

Let $\rho: G \to \GL(V)$ be a representation of $G.$ Then the character is the map $\chi_{\rho}:G \to \setC,$ defined as $\chi_{\rho}(g) \coloneqq \operatorname{tr}(\rho(g)).$

A character is irreducible if it is the character of an irreducible representation.

#Representation theory of $\symS_n$

It is very common to consider $\symS_n$ acting on some vector space. This is then an $\symS_n$-module, and a representation of $\symS_n.$

As a typical example of $\symS_n$-modules, we have the Specht modules.

#Frobenius characteristic

Let $\chi$ be a class function on $\symS_n.$ We then define the Frobenius characteristic (or Frobenius image) as the map \[\frobChar(\chi) \coloneqq \frac{1}{n!} \sum_{\sigma \in \symS_n} \chi(\sigma) \powerSum_\sigma(\xvec) = \sum_{\mu} \chi_\mu \frac{\powerSum_\mu(\xvec)}{z_\mu}.\] This map has the property that for the irreducible characters $\chi^\lambda$ of $\symS_n,$ which are class functions, we have \[\frobChar(\chi^\lambda) = \schurS_\lambda(\xvec).\]

Note that the Hilbert series can be recovered from the Frobenius characteristic. If $M = \bigoplus_k M^k$ is a graded $\symS_n$-module, and \[\frobChar(M) = \sum_{k \geq 0} t^k \sum_{\lambda \vdash n} a(\lambda,k) \schurS_\lambda(\xvec),\] then the Hilbert series is given by \[\sum_{k \geq 0} t^k \sum_{\lambda \vdash n} a(\lambda,k) f^{\lambda},\] where $f^\lambda$ is the number of standard Young tableaux of shape $\lambda.$ The Hilbert series can alternatively be described as $\langle \frobChar(M), \completeH_{1^n} \rangle.$

L. Vanzi studies the irreducible constituents of the Sylow permutation character of $\symS_n$ at the prime $2$ [Van26]. If $n=2^{a_1}+\dotsb+2^{a_t}$ with $a_1\gt{}\dotsb\gt{}a_t\geq 0$ and $n\geq 11,$ then every non-hook partition of $n$ with at most $a_1+t-1$ parts labels such a constituent, and this bound is sharp. He also shows that no constituent has more than $\lceil n/2\rceil$ parts.

S. Brauner, P. Commins, and V. Reiner study invariant theory for the free left-regular band on injective words and for a $q$-analogue with finite general linear group symmetry [BCR23]. Their decompositions use Stirling and $q$-Stirling numbers, random-to-top shuffling, and derangement symmetric functions of D{\'e}sarm{\'e}nien and Wachs.

M. Romero decomposes Kronecker powers of the harmonics of $\symS_n$ by studying the action of $\symS_n$ on polynomial rings with several sets of variables [Rom22]. The combinatorial side uses a generalized comajor statistic, giving generalized principal evaluations for Schur and fundamental quasisymmetric functions.

A. M. Vershik and N. V. Tsilevich study the Schur–Weyl graph, the branching graph attached to Schur–Weyl duality [VT21]. Their graph-theoretic approach gives another route to E. Thoma’s theorem on characters of the infinite symmetric group.

#Computing characters of $\symS_n$

Suppose we have an $\symS_n$-module $M$ and wish to compute its Frobenius characteristic $\frobChar_M(\xvec).$ This gives a step-by-step guide (inspired by lecture notes by M. Zabrocki).

The same method can be adapted to compute characters for arbitrary finite groups.

We assume that $M$ is the span of some set of polynomials in $\setC[x_1,\dotsc,x_n].$
Use linear algebra and find a basis for $M = \langle \bvec_1,\bvec_2,\dotsc,\bvec_d \rangle.$
For each $\sigma \in \symS_n,$ we compute its character $\chi_M(\sigma)$ as follows. For each $i \in [d],$ solve the linear system \[\sigma(\bvec_i) = c_{i1} \bvec_1 + c_{i2} \bvec_2 + \dotsb + c_{id} \bvec_d.\] Then $\chi_M(\sigma) \coloneqq c_{11} + c_{22} + \dotsb + c_{dd}.$ This is a type of trace and independent of the choice of basis.

It is in fact sufficient to compute $\chi_M(\sigma)$ for each type $\mu,$ as $\chi_M(\sigma)=\chi_M(\tau)$ whenever $\sigma$ and $\tau$ have the same cycle type.
Now that we have computed $\chi_M(\sigma),$ we use the above formula and get that \[\frobChar_M(\xvec) = \frac{1}{n!} \sum_{\sigma \in \symS_n} \chi_M(\sigma) \powerSum_\sigma(\xvec).\]

Using the fact that $\frac{n!}{z_\mu}$ is the number of permutations of type $\mu$ and the above observations, we see that \[\frobChar_M(\xvec) = \sum_{\mu \vdash n} \chi_M(\mu) \frac{\powerSum_\mu(\xvec)}{z_\mu},\] where $\chi_M(\mu)$ is the character of some permutation of type $\mu.$
We then have the following relation: \[M = \bigoplus_{\lambda \vdash n} \left(S^{\lambda} \right)^{\oplus c_\lambda} \qquad \Longleftrightarrow \qquad \frobChar_M(\xvec) = \sum_{\lambda \vdash n} c_\lambda \schurS_\lambda(\xvec).\] Here, we have expressed $M$ as a sum of irreducible $\symS_n$-modules, denoted $S^{\lambda}.$ In particular, we must have that the coefficients $c_\lambda$ are non-negative integers. Note that $\dim(S^{\lambda}) = f^\lambda,$ the number of standard Young tableaux of shape $\lambda.$

In all examples below, $\symS_n$ acts on the space by permuting the variable indices.

Example

Let $M = \langle x_1 + x_2 + x_3 \rangle.$ This is a one-dimensional vector space, and all elements in $\symS_3$ fix $M.$ Thus, $\chi_M(\sigma)=1$ for all $\sigma \in \symS_3$ and \[\frobChar_M(\xvec) = \frac{1}{3!} \sum_{\sigma \in \symS_3} \powerSum_\sigma(\xvec) = \schurS_{3}(\xvec).\] The same thing happens for $M = \langle x_1 x_2 x_3 \rangle.$

Example

Let $M = \langle (x_1-x_2)(x_1-x_3)(x_2-x_3) \rangle.$ This is also a one-dimensional vector space, but odd permutations change the sign of the basis vector. Thus, $\chi_M(\sigma)=\sign(\sigma)$ and \[\frobChar_M(\xvec) = \frac{1}{3!} \sum_{\sigma \in \symS_3} \sign(\sigma) \powerSum_\sigma(\xvec) = \schurS_{111}(\xvec).\]

Example

Let $M = \langle x_1 , x_2 , x_3 \rangle.$ Note that $\langle x_1 + x_2 + x_3 \rangle$ is an invariant subspace.

Then $\frobChar_M(\xvec) = \schurS_{3}(\xvec) + \schurS_{21}(\xvec).$

Example

Let $\Delta = \prod_{1\leq i \lt{} j \leq n} (x_j-x_i).$ Let $M$ be the module spanned by $\Delta$ and all partial derivatives of all orders of $\Delta.$ Then the graded Frobenius characteristic is given by \[\frobChar_M(\xvec) = \sum_{\lambda \vdash n} \schurS_\lambda(\xvec) \sum_{T \in SYT(\lambda)} q^{\maj(T)}.\] In particular, $M$ has dimension $n!$; this is a result by M. Haiman.

Example

From [ALW15] we have the following result. The symmetric group $\symS_n$ acts on parking functions by relabeling. We can generalize this to view parking functions as labeled $m$-Dyck paths in an $n \times mn$-rectangle (counted by Fuss–Catalan numbers). The Frobenius characteristic for $(m,n)$ is then \[\sum_{\lambda \vdash n} K_{\lambda,m} \completeH_\lambda\] where $K_{\lambda,m}$ is the number of $m$-Dyck-paths of type $\lambda.$ Orbits of the action above are indexed by $m$-Dyck paths.

Example

For a lecture related to symmetric-group actions on set partitions and non-crossing set partitions, see [Com22]. See also $SL_3$-webs.

Example (Garsia–Procesi modules).

Garsia–Procesi modules, see [CC24].

Example (Garsia–Haiman modules).

Garsia–Haiman modules, see [Arm22].

Example (Involution matrix loci).

J. M. Liu, Y. Ma, B. Rhoades, and H. Zhu apply orbit harmonics to the loci of involution matrices and fixed-point-free involution matrices [LMRZ25]. The resulting graded $\symS_n$-modules have explicit monomial bases and graded Frobenius characteristics. In the fixed-point-free case, the Frobenius image refines the plethysm $\schurS_{n/2}[\schurS_2],$ and the Hilbert series is related to longest decreasing subsequences of fixed-point-free involutions. S. T. Griffin analyzes orbit harmonics for a union of two $\symS_n$-orbits [Gri22]. When the coordinate sums of the two orbits differ, the associated graded representation is a direct sum of two Springer representations, with one shifted in degree by one. S. Kato studies symmetric functions and Springer representations [Kat22], giving further representation-theoretic constructions whose Frobenius characteristics are naturally expressed in symmetric-function bases. H. Zhu applies orbit harmonics to finite matrix loci of rook placements with exactly $r$ rooks [Zhu26]. The resulting graded $\symS_n\times\symS_m$-modules have signed and sign-free graded character formulas, with applications to presentations and module maps. Y. Li, J. Liu, and B. Rhoades apply the same method to the locus of derangement permutation matrices [LLR26]. They give generators for its vanishing ideal and associated graded ideal, a monomial basis, a Hilbert series described through the Foata transformation and longest increasing subsequences, and an alternating-sum formula for the graded $\symS_n$-character. J. Oh and B. Rhoades study quotient rings attached to contingency tables with fixed row and column sums [OR25]. Their standard monomial bases are described by the matrix-ball avatar of RSK, and their Hilbert series are expressed using a zigzag statistic on contingency tables. N. Bergeron, X. Mootoo, and V. Vyas compute the reduced Gröbner basis of a Catalan path ideal [BMV22]. For their quotient ring, the graded Frobenius characteristic is explicitly \[\sum_{k=0}^{\lfloor n/2 \rfloor} \schurS_{(n-k,k)} q^k.\]

#Representation theory of $\GL_n$

We can also study representation theory of infinite groups, such as the group of invertible $n\times n$-matrices with complex entries. This is the general linear group, $\GL_n(\setC).$ For a brief introduction to representation theory of $\GL_n(\setC),$ see for example [Chap. 7, App. 2, Sta01]. For more detailed introductory accounts, see [Kam09] and D. E. Speyer’s blog post on writing down representations of $\GL_n$ [Spe08]. For broader lecture notes on representation theory, see Z. Rosen’s notes from a course by M. Haiman [Ros12] and the notes of P. Etingof et al. [EGHL+11].

A representation of $\GL_n(\setC)$ is a group homomorphism $\phi : \GL_n(\setC) \to \GL_m(\setC)$ for some $m.$ We will only study the cases when $\phi$ is homogeneous and rational, meaning that $\phi(\alpha A) = \alpha^k \phi(A)$ for some $k \in \setZ$ and all $\alpha \in \setC \setminus \{0\}.$ The integer $k$ is called the degree of $\phi,$ and if $k\geq 0,$ we say that $\phi$ is a polynomial representation.

As a concrete example, $\phi : \GL_n(\setC) \to \GL_1(\setC)$ defined as $\phi(A) = \det(A)$ is a homogeneous polynomial representation of degree $n.$

Let $\phi$ be a rational representation of $\GL_n(\setC).$ If $A$ has eigenvalues $x_1,\dotsc,x_n,$ there is a Laurent polynomial \[\glChar(\phi)(\xvec) = \sum_{\alpha \in \setZ^n} m_\alpha x^\alpha\] called the character of $\phi,$ such that $\phi(A)$ has eigenvalue $x^\alpha$ with multiplicity $m_\alpha.$ Moreover, if $\phi$ is polynomial, then $\glChar(\phi)(\xvec)$ is a polynomial in $x_1,\dotsc,x_n.$

Theorem

Every irreducible homogeneous polynomial representation $\phi$ of $\GL_n(\setC)$ is given as \[\glChar(\phi)(\xvec) = \schurS_\lambda(x_1,\dotsc,x_n)\] for some partition $\lambda$ with at most $n$ parts, where $\schurS_\lambda$ is a Schur polynomial.

Given two characters $\phi$ and $\varphi,$ we can define the tensor product $\phi \otimes \varphi.$ We then have \[\glChar(\phi \otimes \varphi)(\xvec) = \glChar(\phi)(\xvec) \cdot \glChar(\varphi)(\xvec).\] In particular, if $\phi$ and $\varphi$ are irreducible, then the character $\glChar(\phi \otimes \varphi)(\xvec)$ decomposes into irreducible representations via the Littlewood–Richardson rule.

For dominant weights $\lambda,\mu$ and a Weyl-group element $w,$ M. Gupta, K.~N.~Raghavan, and S. Viswanath study the Kostant–Kumar module \[\mathcal K(\lambda,w,\mu) =U(\mathfrak g)(v_\lambda\otimes v_{w\mu}).\] These modules interpolate between the Cartan component and the full tensor product. They give presentations, describe the filtration by Demazure projections, and interpret the resulting refined tensor-product multiplicities [GRV26].

If $\phi : \GL(U) \to \GL(V)$ and $\varphi : \GL(V) \to \GL(W)$ are representations, then the composition $\varphi \circ \phi$ is a representation. The character of $\varphi \circ \phi$ is related to the individual characters as \[\glChar(\varphi \circ \phi) = \glChar(\varphi)[ \glChar(\phi) ],\] where we use the plethysm operation.

#Additional topics

For the interaction between representation theory and 0-Hecke algebras, see [HM24].

N. V. Tsilevich, A. M. Vershik, and S. Yuzvinsky associate an intrinsic hyperplane arrangement to an arbitrary irreducible representation of the symmetric group [TVY20].

T. Halverson and T. N. Jacobson construct irreducible modules for the partition algebra and many diagram subalgebras using set-partition tableaux [HJ20]. Their character formulas are expressed as nonnegative sums of symmetric-group characters.

B. Pawlowski, E. Ramos, and B. Rhoades study representation stability for spanning configurations of subspaces in a fixed complex vector space [PRR23]. The spanning line-configuration case is tied to modules arising in the Delta conjecture.

V. Reiner and M. Shimozono define generalized flagged Schur modules and key modules in their work on key polynomials and a flagged Littlewood–Richardson rule [RS95]. In this sense the key and flagged Schur families also have a representation-theoretic model as characters of modules for a Borel subgroup.

In [Sze26], the coordinate rings of general Segre embeddings of products of projective spaces are studied. Their bigraded Hilbert polynomials are expressed in terms of major-descent generating functions of words in multisets. The paper also discusses quantum cohomology and Garsia–Stanton-style bases.

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[Ros12]Zvi Rosen. Representation theory notes. 2012. Notes from Mark Haiman’s Math 252 course

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The symmetric functions catalog

#Representation theory

#Representation of a group

#Irreducible representation

#Tensor products

#Character

#Representation theory of \(\symS_n\)

#Frobenius characteristic

#Computing characters of \(\symS_n\)

#Representation theory of \(\GL_n\)

#Additional topics

Bibliography