Tableaux cyclic sieving

2026-07-09 Improve this page

See the cyclic sieving phenomenon page for the definition and related theorems.

#Standard Young tableaux

#Rectangular SYT

Let $\SYT(a^b)$ be the set of standard Young tableaux with $b$ rows of length $a,$ and let $\langle \partial \rangle = \grpc_{ab}$ be the cyclic group generated by promotion. Finally, for a partition $\lambda \vdash n$ define the $q$-hook formula \[f^{\lambda}(q) \coloneqq \frac{[n]_q!}{\prod_{\square \in \lambda} [h(\square)]_q}.\] Then $(\SYT(a^b), \grpc_{ab}, f^{a^b}(q))$ is a CSP-triple, see B. Rhoades [Rho10].

For a different proof using Wronskians, see [Pur13]. D. Rhee gives an overview of different approaches in his master’s thesis [Rhe12]. See also C. Ahlbach [Ahl19] for some related results and open conjectures. In particular, Ahlbach explicitly describes the fixed points under promotion.

Other alternative proofs can be found in [FK13, Wes16, Wes19].

The $3$-row case can also be seen as rotation of certain graphs introduced in [Kup96]. T. K. Petersen, P. Pylyavskyy, and B. Rhoades prove the cyclic sieving result [PPR08]. These graphs constitute a so-called web basis for $U_q(\mathfrak{sl}_3).$

The $4$-row case can be interpreted as rotation of hourglass plabic graphs introduced in [GPPS+23]. This gives a bijection between $4$-row rectangular SYT and hourglass plabic graphs such that promotion on the SYTs maps to rotation of these planar graphs. These graphs give rise to a web basis for $U_q(\mathfrak{sl}_4).$

#Signed rectangular SYT

S. Pfannerer proves a signed analogue of Rhoades’ rectangular-tableau CSP [Pfa26]. The objects are signed standard tableaux of rectangular shape, and the action combines Schützenberger promotion with cyclic shift of the signs. The sieving polynomial is the super major index generating function introduced by S. Armon and J. P. Swanson. By taking Cartesian products of tableaux, the result also gives cyclic sieving phenomena for arbitrary non-rectangular shapes.

#Rectangular domino tableaux

L. Colmenarejo, B. E. Tenner, and C. E. Thompson prove a cyclic sieving phenomenon for domino tableaux of shape $2\times n$ [CTT26]. Their enumerative formula for these tableaux leads to a new CSP on the $2$-row rectangular case. They also enumerate rectangular domino tableaux in arbitrary dimensions and conjecture a corresponding cyclic sieving phenomenon for general rectangles.

#Stretched SYT

In joint work with S. Pfannerer, M. Rubey, and J. Uhlin, we show that a cyclic group action of order $n$ exists such that for any $\lambda,$ \[\left( \SYT( n \lambda), C_n, f^{n \lambda}(q) \right)\] is a CSP-triple [Thm. 46, APRU20]. The proof is existential, and does not give a natural action on the tableaux.

Problem

Find an explicit order-$n$ action on $\SYT(n\lambda)$ which realizes this cyclic sieving phenomenon for arbitrary $\lambda.$

#Staircase and other SYT

Let $sc_k$ be the staircase shape for the partition $(k,k-1,\dotsc,2,1).$ S. Pon and Q. Wang show that promotion on $\SYT(sc_k)$ acts in a structured way, but they do not prove a cyclic sieving phenomenon [PW11]. Building on his earlier work using Wronskians to understand promotion, and by considering the Lagrangian Grassmannian, K. Purbhoo shows that every promotion symmetry class for the staircase is counted by the number of certain ribbon tableaux [Pur18]. N. Williams conjectured a product formula for a polynomial which should be a CSP polynomial for the action of promotion on $\SYT(sc_k):$ see [Hop20] for this conjecture.

Now let $ssc_k$ be the shifted staircase shape for the strict partition $(k,k-1,\dotsc,2,1).$ By considering the orthogonal Grassmannian, K. Purbhoo shows that every symmetry class of promotion on $\SYT(ssc_k)$ can again be counted by the number of certain ribbon tableaux [Pur18]. Using this result of Purbhoo, Sekheri–Schank–Djermane [SSD14] verified that \[(\SYT(ssc_k),\langle\partial\rangle,f(q))\] is a CSP-triple, where \[f(q)\coloneqq \frac{[k(k+1)/2]_q!}{\prod_{1\leq i \leq j \leq k}[i+j-1]_q},\] which is again the major index generating function for these SYTs.

For more results/conjectures on CSPs for promotion acting on linear extensions of various posets (including the shifted trapezoid shape, the shifted double staircase shape, and the $\mathsf{V}\times[n]$ poset), see [Hop20]. Q. V. Dao, J. Wellman, C. Yost-Wolff, and S. W. Zhang study rowmotion orbits of trapezoid posets [DWYZ20].

#Semistandard type tableaux

#Rectangular SSYT

Let $\SSYT(a^b,k)$ be the set of semistandard Young tableaux with $b$ rows of length $a$ and entries less than or equal to $k.$ Let $\langle \partial \rangle = \grpc_{k}$ be the cyclic group generated by $k$-promotion. Consider the $q$-analog of the hook-content formula: \[X_{\lambda,k}(q) \coloneqq q^{-n(\lambda)} \schurS_\lambda(1,q,q^2,\dotsc,q^{k-1}) = \prod_{(i,j) \in \lambda} \frac{[n+c(i,j)]_q}{ [h(i,j)]_q}.\] Then $(\SSYT(a^b,k), \grpc_{k}, X_{\lambda,k}(q))$ is a CSP-triple, [Rho10]. This result can also be deduced from [SW20] by mapping promotion to toggles.

B. Fontaine and J. Kamnitzer refine B. Rhoades’ result in the following manner [FK13]. Let $\gamma$ be a composition of $ab,$ such that $\gamma$ is invariant under $\ell$-th cyclic shift, and let $\partial$ denote the $k$-promotion operator. Then \[\left( \SSYT(a^b,\gamma), \langle \partial^\ell \rangle, q^{\frac12(a^2 b- (i_1^2+i_2^2+\dotsb + i_m^2) } K_{a^b,\gamma}(q) \right)\] is a CSP-triple. Here $K_{\lambda,\nu}(q)$ is a Kostka–Foulkes polynomial.

#Hook SSYT

M. Bennett, B. Madill, and A. Stokke prove the following results [BMS14]. Let $\alpha$ be a weak composition with $k$ parts and let $\lambda = (n-m,1^m)$ be a hook shape with $n$ boxes. Then \[|\SSYT(\lambda,\alpha)| = \binom{ nz(\alpha)-1}{m}\] where $nz(\alpha)$ is the number of nonzero parts in $\alpha.$

The cyclic symmetry of a weak composition $\alpha$ is defined as the smallest positive integer $p$ such that $\alpha$ is expressible as the concatenation $(\beta,\beta,\dotsc,\beta)$ where $\beta$ is a composition with $p$ parts.

Theorem

Let $\lambda = (n-m,1^m),$ and suppose $\alpha$ is a weak composition of $n$ with $k$ parts and cyclic symmetry $p.$ Furthermore, let $\promotion_k$ denote $k$-promotion. Then \[\left( \SSYT(\lambda,\alpha), \langle \promotion_k^p \rangle, \qbinom{ nz(\alpha)-1}{m}_q \right)\] is a CSP-triple, and the cyclic group has order $nz(\alpha)-1.$

#SSYT of arbitrary shape

Y.-T. Oh and E. Park provide a new CSP on $\SSYT(\lambda,n)$ with a group action of order $n$ whenever $\gcd(n,|\lambda|)=1$ [OP19]. The element $c \coloneqq \cryss_1 \cryss_2 \dotsb \cryss_{n-1} \in \symS_n$ acts on $\SSYT(\lambda,n)$ as a product of crystal reflection operators. Note that this is similar to how the promotion operator is defined, but in contrast with promotion, the element $c$ has order $n$ for all partitions $\lambda$ when acting on $\SSYT(\lambda,n).$ Oh and Park prove that whenever $\gcd(n,|\lambda|)=1,$ \[(\SSYT(\lambda,n), \langle c \rangle, X_{\lambda,n}(q))\] is a CSP-triple, where $X_{\lambda,n}(q) = q^{-n(\lambda)} \schurS_\lambda(1,q,q^2,\dotsc,q^{n-1}).$ They prove that every orbit is free, meaning that all orbits under $c$ have size $n.$ This means that $X_{\lambda,n}(\xi)=0$ for $\xi$ a primitive $n$th root of unity, and that \[X_{\lambda,n}(q) \equiv \frac{|\SSYT(\lambda,n)|}{n} [n]_q \mod (q^n -1).\]

Their result generalizes to skew shapes; see [Ale23], where a shorter proof is given as well.

G. Henrickson, A. Stokke, and M. Wiebe prove a type $C$ analogue for symplectic tableaux [HSW24]. In their setting a product of simple Weyl-group reflections acts on symplectic tableaux, and for suitable odd parameters the orbits are free. The cyclic sieving polynomial is a $q$-analogue of the hook-content formula, and their construction also treats skew symplectic tableaux.

Example (Orbits for $\lambda=32$ and $n=4$).

For example, the following are some of the orbits under $\cryss_{1}\cryss_{2}\cryss_{3}$ on the set $\SSYT(32,4).$ Since $\gcd(4,5)=1,$ the result above applies, and all orbits have size $n=4.$ Each row here is an orbit.

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

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

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

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

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

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

Y.-T. Oh and E. Park prove the non-skew case of the Alexandersson–Amini conjecture [Cor. 3.4, OP21]; see also [AA19].

Theorem (Oh–Park 2020).

Let $\lambda$ be a partition so that $n$ divides $\lambda_i - \lambda_j$ for all $i,j.$ Then a cyclic group $\grpc_n$ of order $n$ exists such that \[(\SSYT(\lambda,m),\grpc_n,q^{-\partitionN(\lambda)}\schurS_{\lambda}(1,q,q^2,\dotsc,q^{m-1}))\] is a CSP-triple.

The cyclic group is not described explicitly. The result by Oh and Park is a semistandard analog of [Thm. 46, APRU20], which concerns standard Young tableaux.

T. Akhmejanov and B. Elek study promotion and cyclic sieving for rectangular $\delta$-semistandard tableaux [AE20]. Their $\delta$-promotion has period $n$ on rectangular tableaux, and the fixed-content cyclic sieving polynomial is a generalized Kostka polynomial.

J. B. Monterrubio, G. Henrickson, and A. Stokke give another semistandard cyclic sieving family [MHS23]. For shapes $\lambda=(m,n^b)$ and $(b+2)$-part contents $\mu,$ they prove a CSP for $\SSYT(\lambda,\mu)$ under a power of jeu-de-taquin promotion, with sieving polynomial given by a shifted modified Kostka–Foulkes polynomial.

S.-Y. Lee and Y.-T. Oh partially resolve the skew case [LO22].

Theorem (Lee–Oh, 2022).

Let $\lambda/\mu$ be a skew shape such that $\lambda_i - \mu_i$ is a multiple of $m$ for all $i.$ Then a cyclic group $\grpc_m$ of order $m$ exists such that \[(\SSYT(\lambda/\mu,km),\grpc_m,\schurS_{\lambda/\mu}(1,q,q^2,\dotsc,q^{km-1}))\] is a CSP-triple, for any $k \in \setN.$

They also prove a similar theorem for ribbons.

Theorem (Lee–Oh, 2022).

Let $\lambda/\mu$ be a ribbon shape, such that $\lambda_i - \mu_i$ is a multiple of $m$ for all $i.$ Then a cyclic group $\grpc_m$ of order $m$ exists such that \[(\SSYT(\lambda/\mu,k),\grpc_m,\schurS_{\lambda/\mu}(1,q,q^2,\dotsc,q^{k-1}))\] is a CSP-triple.

Moreover, Lee and Oh show that the conjecture in [AA19], regarding stretching of general skew shapes, is not true in general. Taking $\lambda/\mu = 3321/21,$ $m=9,$ and $k=4,$ and looking at $\schurS_{9\lambda/9\mu}(1,q,q^2,q^3),$ gives an example where CSP cannot hold [LO22].

The skew shape case is treated further by N. Kumari [Kum22]. Kumari proves root-of-unity factorization results for skew hook Schur functions and interprets the corresponding principal specializations using ribbon supertableaux. For odd periods this gives a cyclic sieving phenomenon on semistandard supertableaux, and the same method also gives a sign-condition generalization of the Lee–Oh theorem for skew semistandard Young tableaux.

#Other types of fillings

#Plane partitions

L. Shen and D. Weng prove the following result using representation theory, cluster variables, and the theory of the Grassmannian [SW20].

Let \[P_{a,b,c}(q) \coloneqq \prod_{i=1}^a \prod_{j=1}^b \prod_{k=1}^c \frac{[i+j+k-1]_q}{[i+j+k-2]_q}.\] This is a $q$-analog of the number of plane partitions that fit in an $a \times b$ rectangle with entries at most $c.$

Let $\pi \in P_{a,b,c},$ and for convenience augment the plane partition with values $\pi_{i,0}=\pi_{0,j}=c$ and $\pi_{a+1,j}=\pi_{i,b+1}=0.$ The linear toggle $\tau_{ij}$ produces a new plane partition $\tau_{ij} \pi$ from $\pi$ as follows: \[(\tau_{ij} \pi)_{kl} = \begin{cases} \pi_{kl} \text{ if } (i,j)\neq (k,l) \\ \max(\pi_{i,j+1},\pi_{i+1,j} ) - \min(\pi_{i-1,j},\pi_{i,j-1} ) - \pi_{i,j} \text{ if } (i,j) = (k,l). \end{cases}\]

Let $\eta$ be the product of linear toggles that hit each square exactly once, from bottom to top, from left to right. That is, \[\eta \coloneqq (\tau_{a,1}\tau_{a-1,1} \dotsm \tau_{1,1})\dotsm (\tau_{a,2}\tau_{a-1,2}\dotsm \tau_{1,2})\dotsm\] Then \[\left(P_{a,b,c}, \langle \eta \rangle, P_{a,b,c}(q) \right)\] is a CSP-triple, where $\eta$ is of order $a+b.$ This extends an earlier result in [RS12], where they consider the case $c=1.$

It turns out that this CSP of Shen–Weng is equivalent to the aforementioned CSP for promotion of rectangular SSYT obtained by B. Rhoades; see [Hop20] for the details of this equivalence.

S. Hopkins considers plane partitions with additional symmetry [Hop20], counting plane partitions in a square under the promotion and transposition-complement operator. The paper contains several interesting conjectures regarding CSP on plane partitions with additional symmetry. See the sections Root posets and Minuscule posets for more discussion on these conjectures.

#Plethysm coefficients

D. B. Rush gives several instances of cyclic sieving related to plethysm coefficients and promotion [Rus18]. For example, if $\lambda$ is a rectangular partition, \[\left( \mathrm{PYTab}(\lambda,\mu^n), \partial, \pm \langle \hallLittlewoodT_{1^n}(\xvec;q) \circ \schurS_\mu, \schurS_\lambda \rangle \right)\] is a CSP-triple. Here, $\mathrm{PYTab}$ is a set of semistandard Young tableaux with given shape and weight, plus a Yamanouchi condition. The function $\hallLittlewoodT_{1^n}(\xvec;q)$ is a transformed Hall–Littlewood polynomial.

#Macdonald $\macdonaldE$ fillings

Consider the specialized nonsymmetric Macdonald polynomial $\macdonaldE_{\lambda}(\xvec;q;0),$ where $\lambda$ is a partition. This makes $\macdonaldE_{\lambda}(\xvec;q;0)$ into a symmetric polynomial: in fact it is more or less a modified Hall–Littlewood polynomial. These polynomials in $k$ variables can be realized as a sum over certain non-attacking fillings, $NAF(\lambda,k).$

In joint work with J. Uhlin, we prove the following results [AU20].

Theorem (Alexandersson, Uhlin 2020).

Let $\lambda$ be an integer partition, and $n$ a positive integer. Then \[\left( NAF(n\lambda,k), \langle \phi \rangle,\macdonaldE_{n\lambda}(\underbrace{1,1,\dotsc,1}_k;q;0) \right)\] is a CSP-triple, where $\phi$ acts by cyclically shifting blocks of $n$ consecutive columns, and redistributing the entries within columns. Moreover, as $n=1,2,3,\dotsc,$ this is a Lyndon-like family.

Furthermore, this can be refined to the case when we let the content $\nu$ be fixed, i.e., \[\left( NAF(n\lambda,\nu), \langle \phi \rangle,[\monomial_\nu]\macdonaldE_{n\lambda}(\xvec;q;0) \right)\] is a CSP-triple for every choice of partition $\nu.$

We prove this theorem by using a result by B. Rhoades [Rho10], regarding cyclic sieving on matrices.

J. Uhlin proves several related results in earlier work [Uhl19]. Note that the case $\lambda = (1)$ gives the CSP with major index on words of length $n$ with entries in $[k].$

#Increasing tableaux

O. Pechenik describes the following CSP [Pec14]. Let $\lambda\vdash n$ and let $Inc_k(\lambda)$ denote the set of tableaux with strictly increasing rows and columns and with maximal value $n-k,$ such that every number in $[n-k]$ is present at least once. Note that $Inc_0(\lambda) = \SYT(\lambda).$

For $T \in Inc_k(2\times n),$ let $\maj(T)$ be the sum of all entries $j$ in row $1,$ such that $j+1$ appears in row $2.$ It is then proved that \[f_{n,k}(q) \coloneqq \sum_{T \in Inc_k(2\times n)} q^{\maj(T)} = q^{n+\binom{k}{2}} \frac{\qbinom{n-1}{k}_q \qbinom{2n-k}{n-k-1}_q }{[n-k]_q}.\] Let $\grpc_{2n-k}$ act by a variant of $k$-promotion on increasing tableaux. Then \[\left( Inc_k(2\times n), \grpc_{2n-k}, f_{n,k}(q) \right)\] is a CSP-triple. Note that $Inc_k(2\times n)$ is in bijection with $\SYT(n-k,n-k,1^k).$

T. Pressey, A. Stokke, and T. Visentin generalize Pechenik’s result and show that \[\left( Inc_k(N-r,1^r), \grpc_{N-k-1}, \qbinom{N-k-1}{r}_q \qbinom{r}{k}_q \right)\] is a CSP-triple, where promotion is used [PSV16].

X. Chen gives formulas for two-row skew shapes; perhaps these exhibit the CSP [Che20].

C. Gaetz, O. Pechenik, J. Striker, and J. P. Swanson prove that packed, increasing tableaux of shape $3 \times k$ and maximal entry $3+k$ are equinumerous with $\SYT(2^3,1^{k-2}).$ They prove that the set of increasing tableaux under $K$-promotion, together with $f^{(2^3,1^{k-2})}(q)$ is a CSP-triple. Interestingly, the promotion has order $k$ here, even though the number of boxes in the SYTs is $k+1$ [GPSS22].

#Alternating sign matrices

There is a canonical $q$-analog of the set of alternating sign matrices, \[ASM(n,q) \coloneqq \prod_{k=0}^{n-1} \frac{(3k+1)_q!}{(n+k)_q!}.\]

N. Stephens-Davidowitz and A. Cloninger attribute a CSP to D. Stanton [N07]. Let $\grpc_4$ act on $ASM(n)$ by a quarter-turn. Then $(ASM(n),\grpc_4,ASM(n,q))$ is a CSP-triple. See also V. Reiner’s lecture on cyclic sieving [Rei18].

Stephens-Davidowitz and Cloninger ask in Question 6.4 whether a map of order $3$ exists such that $ASM(n,q)$ exhibits CSP.

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The symmetric functions catalog

#Standard Young tableaux

#Rectangular SYT

#Signed rectangular SYT

#Rectangular domino tableaux

#Stretched SYT

#Staircase and other SYT

#Semistandard type tableaux

#Rectangular SSYT

#Hook SSYT

#SSYT of arbitrary shape

#Other types of fillings

#Plane partitions

#Plethysm coefficients

#Macdonald \(\macdonaldE\) fillings

#Increasing tableaux

#Alternating sign matrices

Bibliography