The symmetric functions catalog

An overview of symmetric functions and related topics

2021-10-25

Kostka coefficients

The Kostka coefficients appear in the monomial expansion of Schur functions, and the Schur expansion of the complete homogeneous symmetric functions:

\[ \schurS_\lambda = \sum_\mu K_{\lambda\mu} \monomial_\mu, \qquad \completeH_\mu = \sum_\mu K_{\lambda\mu} \schurS_\lambda. \]

Hence, $K_{\lambda\mu}$ is equal to the number of semi-standard Young tableaux of shape $\lambda$ and content $\mu.$

From this, we immediately have that $K_{\lambda\lambda}=1,$ $K_{\lambda,1^n} = f^{\lambda}$ and that $K_{\lambda\mu}=0$ unless $\lambda \trianglerighteq \mu$ in dominance order.

Example (Kostka matrices).

Below is the matrix $(K_{\lambda\mu})$ and its inverse, with rows and columns indexed by the partitions 5, 41, 32, 311, 221, 2111, 11111.

\[ (K_{\lambda\mu})= \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 2 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 & 2 & 3 & 5 \\ 0 & 0 & 0 & 1 & 1 & 3 & 6 \\ 0 & 0 & 0 & 0 & 1 & 2 & 5 \\ 0 & 0 & 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{pmatrix} , \quad (K_{\lambda\mu})^{-1}= \begin{pmatrix} 1 & -1 & 0 & 1 & 0 & -1 & 1 \\ 0 & 1 & -1 & -1 & 1 & 1 & -2 \\ 0 & 0 & 1 & -1 & -1 & 2 & -2 \\ 0 & 0 & 0 & 1 & -1 & -1 & 3 \\ 0 & 0 & 0 & 0 & 1 & -2 & 3 \\ 0 & 0 & 0 & 0 & 0 & 1 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{pmatrix} \]

See the overview of transition matrices for a combinatorial interpretation of the entries in the inverse matrix.

Recursions for Kostka coefficients

From the definition of Kostka coefficients, there is a straightforward recursion. This recursion is equivalent to the Pieri rule. Let $\mu = (\mu_1,\dotsc,\mu_\ell)$ be a partition with $\ell \geq 2$ parts, and let $\mu^{(\ell)}$ denote the partition $(\mu_1,\dotsc,\mu_{\ell-1}),$ that is, with last part removed. Then

\[ K_{\lambda \mu} = \sum_{\nu} K_{\lambda,\mu^{(\ell)}} \]

where the sum is taken over all partitions $\nu$ of size $|\lambda|-\mu_\ell,$ such that $\lambda/\nu$ is a horizontal strip.

See also the recursion for Kostka–Foulkes polynomials.

In [p.327, Mac95], the following recursion is described. In fact, it generalizes to Jack polynomials.

Theorem.

First, let $d(\lambda)\coloneqq \partitionN(\lambda')-\partitionN(\lambda).$ Let $\mu$ be a partition of length $\ell.$ Then we have the recursion

\[ K_{\lambda, \mu} = \frac{1}{ d(\lambda) - d(\mu)} \sum_{1 \leq i \lt j \leq \ell} \sum_{1 \leq r \leq \mu_j} (\mu_i - \mu_j) K_{\lambda, (\mu+r \evec_i - r \evec_j)^*} \]

where $(\mu+r \evec_i - r \evec_j)^*$ denotes the partition obtained from $\mu$ by decreasing entry $i$ by $r,$ and increasing entry $j$ by $r,$ and rearrange in decreasing fashion. The initial conditions are given by $K_{\mu, \mu}=1,$ and $K_{\lambda, \mu}=0$ unless $\lambda \geq \mu$ in dominance order.

Recursions for inverse Kostka coefficients

A convenient recursive formula for the entries in $(K_{\lambda\mu})^{-1}$ is given in [ER90].

Theorem (Eğecioğlu–Remmel, 1990).

Let $\lambda = (r_1^{j_1}, r_2^{j_2},\dotsc,r_k^{j_k})$ where $r_1 \lt r_2 \lt \dotsb.$ Let $\mu = (\mu_1,\dotsc,\mu_\ell)$ in increasing order. We have that

\[ K^{-1}_{\lambda\mu} = \sum_{\substack{1 \leq j \leq d \\ r_j = \mu_i+j-1}} (-1)^{i-1} K^{-1}_{\lambda[j], (\mu_1-1,\mu_2-1,\dotsc,\mu_{i-1}-1,\mu_{i+1},\dotsc,\mu_{\ell})} \]

where $\lambda[j]$ denotes the partition obtained from $\lambda$ with one part equal to $j$ removed.

An implementation of this recursion in Mathematica can be found in this github file.

In [Car98], a $t$-version of the Eğecioğlu–Remmel result is proved, where $(K_{\lambda\mu}(t))^{-1}$ is given a combinatorial interpretation.

A similar-looking but different recursion is given in [Thm. 3, Dua03].

Theorem (H. Duan, 2003).

Let $\lambda = (r_1^{j_1}, r_2^{j_2},\dotsc,r_k^{j_k})$ where $r_1 \lt r_2 \lt \dotsb.$ Let $\mu = (\mu_1,\dotsc,\mu_\ell)$ in increasing order. We have that

\[ K^{-1}_{\lambda\mu} = \sum_{r_j \geq \mu_\ell} (-1)^{r_j-\mu_\ell} \sum_{\nu} K^{-1}_{\lambda[j], \nu} \]

where $\lambda[j]$ denotes the partition obtained from $\lambda$ with one part equal to $j$ removed, and the inner sum is taken over all $\nu$ such that $\mu/\nu$ is a vertical strip of size $r_j-\mu_\ell.$

See also [WZ18], where the coefficients appearing in

\[ \schurS_\mu \hallLittlewoodP_\nu(t) = \sum_\lambda \bar{K}^\lambda_{\mu\nu}(t) \schurS_{\nu} \]

are studied.

The Kostant partition function

See Harris, Insko and Omar [HIO18] for a short introduction.

Let $\evec_{ij}$ denote the vector with $1$ at index $i$ and $-1$ at index $j$ and remaining entries equal to $0.$ We refer to this collection of vectors as positive roots or just roots for short. Given an integer vector $w$ of length $n,$ let Kostant's partition function $\kostantP(w)$ be the number of solutions to

\[ w = \sum_{(i,j) \in \binom{[n]}{2}} a_{ij} \evec_{ij} \]

with $a_{ij}\geq 0$ being non-negative integers. This is the definition in root system type $A_n$ — there are analogues of the partition function in other types. Clearly $\kostantP(w)=0$ unless $w_1+w_2+\dotsb+w_n=0.$

There is a $q$-analogue of $\kostantP(w),$ denoted $\kostantP_q(w)$ introduced by Lustzig [Lus83]. It is defined as

\[ \kostantP_q(w) \coloneqq \sum_{ (a_1,\dotsc,a_\ell) } q^{a_1+\dotsb + a_\ell} \]

where we sum over all non-negative solutions $(a_1,\dotsc,a_\ell)$ that expresses $w$ as a non-negative linear combination of roots.

From the definition above, we have that

\[ \prod_{1 \leq i \lt j \leq n} \left( 1 - q\frac{x_i}{x_j} \right)^{-1} = \sum_{ w \in \setZ^n } \kostantP_q(w) \xvec^w. \]

This has a close relationship with the Jacobi–Trudi identity, see the excellent introduction in [DLT94].

Example.

Let $w=(5,2,-7),$ and our roots are $\evec_{12},$ $\evec_{13}$ and $\evec_{23}.$ We have that $\kostantP(w) = 6,$ since there are $6$ solutions with non-negative coefficients:

\[ 052, 143, 234, 325, 416, 507 \]

For example, $(5,2,-7) = 3\evec_{12} + 2\evec_{13} + 5\evec_{23}.$ Furthermore,

\[ \kostantP_q(w) = q^{12}+q^{11}+q^{10}+q^9+q^8+q^7. \]
Mathematica code

KostantPartitionFunction[w_List] /; Tr[w] != 0 = {};
KostantPartitionFunction[w_List] := Module[
   {a, n = Length[w], roots, idx, vars, sol},
   idx = Subsets[Range[n], {2}];
   roots = Table[SparseArray[ss->{1,-1},{n}], {ss,idx}];
   vars = a /@ idx;
   sol = Solve[
     And @@ Join[Thread[vars >= 0],
       Thread[w == vars.roots]]
     , vars, Integers];
   If[Length[sol] == 0, {}, vars /. sol]
];
KostantPartitionFunction[{5,2,-7}]

Kostant's partition function is related to juggling sequences, see [BHHMS20].

The number of summands used in expressing $w,$converges to a Gaussan distribution, see [HRS19]. Similar properties for other root systems hold.

Kostant partition function in general types

In more generality, let $\Phi^+ \subset \Phi$ be a set of positive roots in a root system. The Kostant partition function for $\Phi^+$ is defined via the relation

\begin{equation*} \prod_{\alpha \in \Phi^+} \frac{1}{1-e^\alpha} = \sum_{w} \kostantP(w) e^w. \end{equation*}

Note that $e$ is just a formal variable in this context, but with this convention, $e^{\alpha} e^{\beta} = e^{\alpha + \beta}.$ The $q$-analogue is then defined via

\begin{equation*} \prod_{\alpha \in \Phi^+} \frac{1}{1-q e^\alpha} = \sum_{w} \kostantP_q(w) e^w. \end{equation*}

Thus, the coefficient $[q^j]\kostantP_q(w)$ count the number of ways to express $w$ as a sum of exactly $j$ positive roots.

In [Ras04b], a different $q$-analogue, defined via

\begin{equation*} \prod_{\alpha \in \Phi^+} \frac{1+(q-1)e^\alpha}{1- e^\alpha} = \sum_{w} \kostantP^*(w;q) e^w, \end{equation*}

is considered. Then $[q^j]\kostantP^*(w;q)$ is the number ways to express $w$ as a sum of positive roots, where exactly $j$ roots appear with a positive multiplicity.

See http://de.arxiv.org/pdf/1111.6648.pdf for a connection with Fibonacci numbers.

Kostka coefficients from Kostant's partition function

Let $\lambda$ and $\mu$ be integer partitions of $n.$ Then

\[ K_{\lambda,\mu} = \sum_{\sigma \in \symS_n} (-1)^{\length(\sigma)} \kostantP\left( \sigma(\lambda + \rho) - (\mu+\rho) \right) \]

where $\rho = (n-1,n-2,\dotsc,2,1,0).$ However, this is a really inefficient way to compute Kostka coefficients.

The Kostka–Foulkes polynomials $K_{\lambda,\mu}(q)$ can be computed via the $q$-analogue:

\[ K_{\lambda,\mu}(q) = \sum_{\sigma \in \symS_n} (-1)^{\length(\sigma)} \kostantP_q\left( \sigma(\lambda + \rho) - (\mu+\rho) \right). \]
Theorem (See [DLT94]).

We have that

\[ \completeH_\mu(\xvec) = \sum_{\alpha \in \setZ^n} \kostantP(\alpha)\schurS_{\mu + \alpha}(\xvec). \]

Littlewood–Richardson coefficients from the partition function

Let $\lambda,$ $\mu$ and $\nu$ be partitions of length at most $k,$ and such that $|\lambda|+|\mu| = |\nu|.$ Set

\[ \delta \coloneqq \frac{1}{2}(k-1,k-3,\dotsc,3-k,1-k). \]
Theorem (Steinbergs formula [Ste61Ras04b]).

We have that the Littlewood–Richardson coefficient $c^{\nu}_{\lambda \mu}$ is given by

\[ c^{\nu}_{\lambda \mu} = \sum_{\sigma, \tau \in \symS_k} (-1)^{\inv(\sigma \tau)} \cdot \kostantP\left( \sigma(\lambda+\delta) + \tau(\mu+\delta) - (\nu+2\delta) \right). \]
Mathematica code

LittlewoodRichardson[lam_List, mu_List, nu_List] := 
  Module[{perms, nlam, nmu, nnu, k, delta},
   k = Max[Length /@ {lam, mu, nu}];
   perms = Permutations@Range[k];
   {nlam, nmu, nnu} = PadRight[#, k] & /@ {lam, mu, nu};
   delta = 1/2 Table[j, {j, k - 1, -(k - 1), -2}];
   Sum[
    With[{
      vec = (nlam + delta)[[sigma]]
        +
        (nmu + delta)[[tau]]
        -
        (nnu + 2 delta)},
     (-1)^Inversions[sigma[[tau]]]
      Length[KonstantPartitionFunction[vec]]
     ]
    , {sigma, perms}, {tau, perms}]
   ];

(* This should return 1 *) LittlewoodRichardson[{2, 2}, {3}, {4, 2, 1}] (* This should return 2 *) LittlewoodRichardson[{2, 1, 1}, {3, 1}, {4, 2, 1, 1}]

A similar expression for the Schur P structure coefficients can be found in [Thm. 6.2.5, Ras04b]. This can perhaps be seen as a type B analog of Steinbergs formula.

Kostka–Foulkes polynomials from charge

For an excellent background on charge, see [Chapter 2.4, But94].

Here is a longer explanation on how to compute the coefficients $K_{\lambda,\mu}(q),$ the Kostka–Foulkes polynomials by using charge. These coefficients appear in the expansion of Schur polynomials into Hall–Littlewood polynomials. The combinatorial model using charge was first described by Lascoux and Schützenberger in [LS78].

First we need to introduce the charge of a permutation. For $\sigma \in \symS_k,$ we let

\begin{equation*} \charge(\sigma) \coloneqq \maj(\rev(\sigma^{-1})) = \sum_{i \notin \DES(\sigma^{-1})} (k-i). \end{equation*}

For example,

\[ \charge(198423765) = \maj(\rev(156498732) ) = \maj(237894651) = 20. \]

The cocharge of a permutation, $\cocharge(\pi)$ is defined as $\binom{n}{2}-\charge(\pi).$

Example (Computing cocharge).

Computing $\cocharge(198423765)$ can be done as follows. Write the permutation and write a 0 as a subscript for $1.$

\[ 1_0\; 9\; 8\; 4\; 2\; 3\; 7\; 6\; 5 \]

We then move to the next larger element in the permutation with no subscript. If moving to the left, the subscript is increased by one, otherwise it stays the same. The first two steps here is to the right, so the subscripts are set to $0.$

\[ 1_0\; 9\; 8\; 4\; 2_0\; 3_0\; 7\; 6\; 5 \]

We then move to the left,

\[ 1_0\; 9\; 8\; 4_1\; 2_0\; 3_0\; 7\; 6\; 5, \]

and then to the right:

\[ 1_0\; 9\; 8\; 4_1\; 2_0\; 3_0\; 7\; 6\; 5_1. \]

Proceeding in this fashion, we arrive at

\[ 1_0\; 9_5\; 8_4\; 4_1\; 2_0\; 3_0\; 7_3\; 6_2\; 5_1. \]

The cocharge is now the sum of the subscripts, $0+5+4+1+0+0+3+2+1 = 16.$ This agrees with the fact that $16+20 = 36 = \binom{9}{2}.$

Standard subwords

Given a word $w$ with content $\mu \vdash n,$ partition its entries into standard subwords as follows. Start from the right of $w$ and mark the first occurrence of $1.$ Proceed to the left and mark the first occurrence of $2,$ then $3$ and so on, wrapping around the end if nessecary, until $\mu'_1$ entries have been marked. This subword is the first standard subword of $w.$ Remove this subword, and repeat the process to find the second standard subword, of length $\mu'_2.$

For example, the first standard subword in $w = 2 1 1 2 3 5 4 3 4 1 1 2 2 3$ has been underlined.

\[ 2, 1, 1, \underline{2}, 3, \underline{5}, 4, 3, \underline{4}, 1, \underline{1}, 2, 2, \underline{3}. \]

In total, we have four standard subwords in $w,$ with corresponding charge values

\[ \charge(25413) = 3,\quad \charge(2431)=2 \quad \charge(132)=2 \quad \charge(12) = 1, \]

and we define $\charge(w)$ as the sum of the charge values of the standard subwords. Hence $\charge(w) = 8$ for our particular word.

Charge of SSYT

For a semistandard Young tableau $T \in \SSYT(\lambda,\mu),$ the reading word of $T,$ $\rw(T)$ is given by concatenating the rows of $T,$ starting with the bottom-most row. We then define $\charge(T) = \charge(\rw(T)),$ and the Kostka–Foulkes polynomial $K_{\lambda,\mu}(q)$ may be computed as

\[ K_{\lambda,\mu}(q) = \sum_{T \in \SSYT(\lambda,\mu)} q^{\charge(T)}. \]

Similarly, the modified Kostka–Foulkes polynomials are defined as

\[ \tilde{K}_{\lambda,\mu}(q) = \sum_{T \in \SSYT(\lambda,\mu)} q^{\cocharge(T)}. \]

We have the relation

\[ \tilde{K}_{\lambda,\mu}(q) = q^{\partitionN(\mu)}$ K_{\lambda,\mu}(q^{-1}). \]
Example (Computing a Kostka–Foulkes polynomial).

We shall now compute $K_{421,3211}(q).$ There are four tableaux in $\SSYT(421,3211)$:

$1$$1$$1$$4$
$2$$2$  
$3$   
$1$$1$$1$$3$
$2$$2$  
$4$   
$1$$1$$1$$2$
$2$$4$  
$3$   
$1$$1$$1$$2$
$2$$3$  
$4$   

The corresponding standard subwords and charge values are given by

\begin{equation*} \underset{1+0+0}{ 3214,\; 21,\; 1 } \qquad % \underset{2+0+0}{4213,\; 21,\; 1 } \qquad % \underset{1+1+0}{ 3241,\; 12,\; 1} \qquad % \underset{2+1+0}{4231,\; 12,\; 1} \end{equation*}

Hence, $K_{\lambda\mu}(q) = q+2q^2+q^3.$

Example (Table of $K_{\lambda,\mu}(q),$ for size $4$).

Here are all $K_{\lambda,\mu}(q)$ for partitions of size $4.$ The rows are indexed by $\lambda.$

$\;$$\textbf{4}$$\textbf{31}$$\textbf{22}$$\textbf{211}$$\textbf{1111}$
$\textbf{4}$$1$$0$$0$$0$$0$
$\textbf{31}$$q$$1$$0$$0$$0$
$\textbf{22}$$q^2$$q$$1$$0$$0$
$\textbf{211}$$q^3$$q^2+q$$q$$1$$0$
$\textbf{1111}$$q^6$$q^5+q^4+q^3$$q^4+q^2$$q^3+q^2+q$$1$

Properties of charge

The elementary Knuth transforms are transformations on three adjacent letters on words. We allow the transformations $yzx \leftrightarrows yxz $ whenever $x \lt y\leq z$ and $xzy \leftrightarrows zxy$ whenever $x \leq y \lt z.$ Two words are Knuth-equivalent, denoted $\equiv$ if one can obtain one from the other via a sequence of elementary Knuth transforms. See also the close relationship with the Robinson–Schensted–Knuth correspondence.

For words of partition weight, we have some extra nice properties: Each equivalence class contains a unique word which is the reading word of a semi-standard Young tableau. Furthermore, one can show that $w \equiv w'$ then $\charge(w)=\charge(w'),$ see [Cor. 2.4.38, But94].

Theorem (Proved in [LS78]).

Charge on words is the unique statistic with the following properties:

  • $\charge(\emptyset) = 0,$
  • $\charge(w) = \charge(\sigma\cdot w),$ where $\sigma$ act via Lascoux–Schützenberger involutions,
  • if $x \neq 1$ and $wx$ has partition-content, then $\charge(wx) = \charge(xw)+1,$
  • if $w1^m$ has partition-content and $w$ contains no ones, then $\charge(w1^m) = \charge(w),$
  • if $w$ and $w'$ are Knuth-equivalent, then $\charge(w) = \charge(w').$

Skew Kostka–Foulkes polynomials

The skew Kostka–Foulkes polynomials $K_{\lambda/\mu,\nu}(q)$ are defined as the coefficients in the expansion

\[ \schurS_{\lambda/\mu}(\xvec) = \sum_{\nu}K_{\lambda/\mu,\nu}(q) \hallLittlewoodP_{\nu}(\xvec;q). \]

One can show that the above formula generalizes, see [Corr. 2.5.8, But94].

\[ K_{\lambda/\mu,\nu}(q) = \sum_{T \in \SSYT(\lambda/\mu,\nu)} q^{\charge(T)}. \]

One can also use rigged configurations to compute Kostka–Foulkes polynomials in an efficient manner.

Cyclage

In [LS81], Lascoux and Schützenberger introduce the notion of cyclage. Let $\mu$ be a partition, and consider the set $\SSYT(\cdot,\mu) \coloneqq \bigcup_\lambda \SSYT(\lambda,\mu).$ That is, the set of all semistandard Young tableaux with content $\mu.$ They define a poset structure on $\SSYT(\cdot,\mu)$ via the covering relations

\[ T \prec T' \text{ if there is $i\geq 2$ such that } \rw(T)\cdot i \equiv i\cdot \rw(T'), \]

where $\equiv$ denotes Knuth equivalence. This is in fact a graded poset, whose grading is given by cocharge. See [Section 5.6, Lot02] for more properties of cyclage.

Example (Figure of a cyclage poset).

The charge is displayed on the right hand side of each tableau.

Cyclage poset

Catabolism

Catabolism is defined in [Exercise 5.6.1, Lot02] as follows. Given a tableau $T$ of shape $\lambda,$ let $F$ be the first row of length $\lambda_1$ and $R$ be the tableau consisting of the remaining rows. Perform row insertion on $F \cdot \rw(R).$ This gives a new tableau with the same weight as $T,$ and charge has increased by $|\lambda|-\lambda_1.$ Tableaux with only one row are the only fixed points under catabolism.

Example (Catabolism of a tableau).

In this example, we start with

$1$$1$
$2$$3$
$4$ 

The top row is moved to the bottom, to create a skew shape.

  $2$$3$
  $4$ 
$1$$1$  

We then perform jeu-de-taquin to get a straight shape.

  $2$$3$
  $4$ 
$1$$1$  
  $2$$3$
 $1$$4$ 
$1$   
  $2$$3$
$1$$1$$4$ 
 $1$$2$$3$
$1$ $4$ 
$1$$1$$2$$3$
  $4$ 
$1$$1$$2$$3$
$4$   

The final tableau is the result of catabolism. This procedure is equivalent with the one described above.

Example (Figure of catabolism (poset)).

The charge is displayed on the right hand side of each tableau.

Catabolism

Identities

Using the Robinson–Schensted–Knuth correspondence, one can show that

\[ \sum_{\lambda \vdash n} K_{\lambda \mu} K_{\lambda \nu} = N_{\mu\nu}, \]

where $N_{\mu\nu}$ is the number of $n \times n$-matrices with entries in $\setN,$ row sums are determined by $\mu$ and column-sums are determined by $\nu.$

There is a $q$-analogue of this in the case $\nu=1^n,$ see [Corolloary 1.3, Kir00].

\[ \sum_{\lambda \vdash n} K_{\lambda \mu} K_{\lambda, 1^n}(q) = q^{n(\mu')} \qbinom{n}{\mu_1,\dotsc,\mu_\ell}_q. \]
Theorem (See [BJ16]).

For a partition $\lambda$ let

\[ \lambda^{(i)} \coloneqq (\lambda_1+1,\lambda_2+1,\dotsc,\lambda_{i-1}+1,\lambda_{i+1},\dotsc,\lambda_{\ell}). \]

Furthermore, $\lambda^{[i]}\coloneqq (\lambda_{i+1},\dotsc,\lambda_\ell).$ With these definitions, we have the following recursion for Kostka–Foulkes polynomials:

\[ K_{\lambda\mu}(q) = \sum_{i=1}^{\mu_1} (-1)^{i-1} q^{\lambda_1-\mu_1-i+1} \sum_{\tau^i} K_{\tau^i,\mu^{[1]}}(q) \]

where the sum runs over all partitions $\tau^i$ such that $\tau^i/\lambda^{(i)}$ are horizontal $(\lambda_1-\mu_1-i+1)$-strips.

Parabolic Kostka polynomials

The parabolic Kostka polynomials $K_{\lambda,R}(q)$ (or generalized Kostka polynomials) generalize the Kostka–Foulkes polynomials. The following definition appear in [SW00].

Let $R = (R^1,\dotsc,R^k)$ be a sequence of partitions, and let $\lambda$ be a partition such that $|\lambda| = \sum_j |R^j|.$ Let $\gamma$ be the partition whose parts is the multiset of parts of the $R^j.$ Furthermore, let $\eta_j \coloneqq \length(R^j)$ and $N \coloneqq \sum_j \eta_j.$ Let $M$ be the $N\times N$ matrix with diagonal blocks of size $\eta_j,$ and let $R^+_\eta \subset [N]^2$ be the subset of entries in $M$ strictly above the blocks.

Example (From this paper).

Let $R=((4,4),(3),(1,1,1)).$ Then $\eta = (2,1,3),$ $N=6$ and $\gamma = 443111.$ The matrix $M$ is as follows where $R^+_\eta$ consists of the lightgreen boxes.

      
      
      
      
      
      

Hence,

\[ R^+_\eta = \{13,14,15,16, \; 23,24,25,26, \; 34, 35, 36 \}. \]

Introduce $\kostantP^\eta_q(w),$ which is a generalization of the $q$-analogue of the Kostant partition function, via the relation

\begin{equation*} \sum_{w \in \setZ^n} \kostantP^\eta_q(w) = \prod_{(i,j) \in R^+_\eta} \frac{1}{1-q\frac{x_i}{x_j}}. \end{equation*}

Finally, the parabolic Kostka polynomial $K_{\lambda,R}(q)$ is defined as

\begin{equation*} K_{\lambda,R}(q) = \sum_{\sigma \in \symS_n} (-1)^{\length(\sigma)} \kostantP^\eta_q\left( \sigma(\lambda + \rho) - (\mu+\rho) \right), \end{equation*}

where $\rho = (N-1,\dotsc,2,1,0).$ It is straightforward from this definition to see that if each partition $R_i = (\mu_i),$ a single part, then $K_{\lambda,R}(q) = K_{\lambda,\mu}(q),$ the standard Kostka–Foulkes polynomial.

One can show [Prop. 8, SW00] that

\[ K_{\lambda,R}(1) = \langle \schurS_\lambda, \schurS_{R^1} \schurS_{R^2} \dotsm \schurS_{R^k} \rangle, \]

which is a type of generalized Littlewood–Richardson coefficient.

Rectangular special case

When all $R^j$ are rectangles, the $K_{\lambda,R}(q)$ are Poincaré polynomials.

Several conjecture regarding this special case appear in these notes. In particular, there is a connection with LLT polynomials.

When the part sizes in the rectangles $R^j$ are weakly decreasing, we have a dominant sequence of rectangles. In this case, it is knonw that $K_{\lambda,R}(q) \in \setN[q],$ via a rigged configuration proof. A charge formula is conjectured in [Conj. 27, SW00], and later proved by M. Shimozono [Thm. 11, Shi01]. There,

\[ K_{\lambda,R}(q) = \sum_{T \in LRT(\lambda,R)} q^{\charge_R(T)} \]

where $\charge_R$ is a generalization of charge, and $LRT(\lambda,R)$ is the set of certain Littlewood–Richardson tableaux. See [KS02KSS01SW99] for more background.

Atomic decomposition

http://fpsac2019.fmf.uni-lj.si/resources/Posters/76poster.pdf

References