#Petrie symmetric functions
The Petrie symmetric functions are a family of symmetric functions indexed by two nonnegative integers. This family is the main topic in the work of D. Grinberg [Gri20, Gri20]. They are also studied independently by H. Fu and Z. Mei [FM20] under the name truncated homogeneous symmetric functions.
An early reference is [Eq. (3.11), DW92], under the name modular complete symmetric functions. A corresponding family of modular elementary symmetric functions is also defined.
A later paper by A. Bazeniar, M. Ahmia, and H. Belbachir studies the Petrie symmetric functions under the name generalized elementary symmetric functions, while studying variants of \(q\)-binomial coefficients [BAB18]. See also M. Ahmia and M. Merca [AM20].
The Petrie symmetric functions are special cases of the modular Schur functions.
The Petrie symmetric function \(G(k,m)\) is defined as \[G(k,m) \coloneqq \sum_{\alpha : |\alpha|=m, \max(\alpha) \lt k} \xvec^{\alpha} = \sum_{\lambda \vdash m, \lambda_1 \lt k} \monomial_\lambda(\xvec)\] where the first sum is over weak compositions \(\alpha\) of \(m\) whose entries are all less than \(k.\) Note that for \(k=2,\) we have \(G(2,m)=\elementaryE_m.\) Hence, the alternative notation \(E^{(k)}_m(\xvec)\) used in [BAB18, AM20] is natural. There is a shift in indexing: in their notation the elementary symmetric functions are recovered at \(k=1.\)
Note that \(G(k,m)\) is homogeneous of degree \(m.\) The nonhomogeneous version is defined as \[G(k) \coloneqq \sum_{m \geq 0} G(k,m) = \sum_{\lambda : \lambda_1 \lt k} \monomial_\lambda(\xvec),\] where the second sum is over all integer partitions with all parts less than \(k.\)
The Petrie symmetric functions also appear in [Num07] as a special case of weighted complete homogeneous symmetric polynomials, \(\completeH^{(a)}_i(x_1,\dotsc,x_n)\) indexed by an integer \(i\geq 0\) and a vector \(a.\) We then let \(a(t)\coloneqq \sum_{i\geq 0} a_i t^i\) and set \[\completeH^{\{a\}}_i(x_1,\dotsc,x_n) \coloneqq [t^i] a(t x_1) a(t x_2)\dotsm a(t x_n).\] This is a symmetric polynomial. Now we can see that \(G(k,m)=\completeH^{\{a\}}_m(\xvec)\) for the sequence \(a_i=1\) if \(i \lt k\) and \(0\) otherwise.
#Recurrence relations and identities
For this section, see [BAB18]. Define \(E^{(s)}_k(\xvec)\) and \(H^{(s)}_k(\xvec)\) via \[\sum_{k=0}^\infty H^{(s)}_k(x_1,\dotsc,x_n) t^k = \prod_{i=1}^n \left( 1-x_i t + \dotsb + (-x_i t)^s \right)^{-1},\] and \[\sum_{k=0}^\infty E^{(s)}_k(x_1,\dotsc,x_n) t^k = \prod_{i=1}^n \left( 1 + x_i t + \dotsb + (x_i t)^s \right).\] Note that \(E^{(s)}_k(\xvec)\) is equal to \(G(s+1,k).\)
\[E^{(s)}_k(x_1,\dotsc,x_n) = \sum_{j=0}^s x_n^j E^{(s)}_{k-j}(x_1,\dotsc,x_{n-1})\] A similar recursion is given for \[H^{(s)}_k(x_1,\dotsc,x_n) = \sum_{j=0}^s (-1)^j x_n^j H^{(s)}_{k-j}(x_1,\dotsc,x_{n-1}).\]
In [AM20] the following generalized Newton identity is proved, where \(s,k\) and \(n\) are positive integers: \[\sum_{j=0}^k (-1)^j E^{(s)}_j(x_1,\dotsc,x_n) H^{(s)}_{k-j}(x_1,\dotsc,x_n) = \delta_{k,0}.\]
M. Ahmia and M. Merca also give relations involving evaluations of monomial symmetric functions at roots of unity [AM20].
#Combinatorial interpretation
There is a combinatorial interpretation of the \(E^{(s)}_j(x_1,\dotsc,x_n)\) in terms of lattice paths.
#Principal specialization
We have [Cor. 4.1, AM20] that \[E^{(s-1)}_k(1,q,q^2,\dotsc,q^{n-1}) = \sum_{j=0}^{\lfloor k/s \rfloor} (-1)^j q^{s \binom{j}{2}} \qbinom{n}{j}_{q^s} \qbinom{n+k-sj-1}{k-sj}_q.\]
#Schur expansion
The Petrie symmetric functions expand with coefficients in \(\{-1,0,1\}\) in the Schur basis: \[G(k) = \sum_\lambda H_k[ \schurS_\lambda ] \schurS_\lambda(\xvec)\] where \(H_k\) is an algebra homomorphism defined via \[H_k[\completeH_j] \coloneqq \begin{cases} 1, & \text{if } 0 \leq j \lt k, \\ 0, & \text{otherwise}. \end{cases}\] Note that the Jacobi–Trudi identity for Schur functions implies that \(H_k[\schurS_\lambda]\) is the determinant of a Petrie matrix, which always has value \(\pm 1\) or \(0\); see [GW74] for background.
Example (Schur expansion of \(G(4,8)\)).
The Schur expansion of \(G(4,8)\) is \[\begin{aligned} G(4,8) = {}& \monomial_{332}+\monomial_{2222}+\monomial_{3221} +\monomial_{3311}+\monomial_{22211}\\ &+\monomial_{32111}+\monomial_{221111} +\monomial_{311111}+\monomial_{2111111} +\monomial_{11111111} \end{aligned}\] and this has Schur expansion \[\schurS_{332}+\schurS_{2222}-\schurS_{3221} +\schurS_{311111}-\schurS_{2111111}+\schurS_{11111111}.\]
An explicit formula for the coefficients in the Schur expansion is proved in [Gri20]. A more explicit formula using \(k\)-cores is proved in [CCEF+22]. These formulas are used to prove the Liu–Polo conjecture [Rem. 1.4.5, LP21].
#Nabla and Schur positivity
The Petrie symmetric functions have a clean sign rule under the \(\nabla\) operator. The following theorem was proved by M. Qu [Qu26].
Theorem (Qu).
Let \(k\geq 2,\) \(n\geq 1,\) and \(r\geq 1.\) Then \[\begin{cases} (-1)^n \nabla^r G(k,n), & \text{if } k\mid n,\\ (-1)^{n-1} \nabla^r G(k,n), & \text{if } k\nmid n, \end{cases}\] is Schur-positive.
This resolves the sign-pattern problem for \(\nabla^r G(k,n)\) posed by F. Bergeron and recorded by D. Grinberg [Conj. 5.2, Gri20].
Example (Sign examples for \(\nabla^r G(k,n)\)).
Since \(G(2,n)=\elementaryE_n,\) the case \(k=2\) recovers the usual Schur positivity of \(\nabla^r\elementaryE_n.\) In degree \(2,\) \[\nabla G(2,2) = \nabla \elementaryE_2 = \schurS_2 + (q+t)\schurS_{11}.\] For \(k=3,\) the same theorem says, for example, that \(-\nabla^r G(3,3)\) is Schur-positive, while \(\nabla^r G(3,5)\) is Schur-positive.
The Petrie symmetric functions have the following property: For every \(\lambda,\) consider the expansion \[\schurS_\lambda \cdot G(k) = \sum_\mu \epsilon(\lambda,k,\mu) \schurS_\mu.\] The coefficients \(\epsilon(\lambda,k,\mu)\) then have the property that they lie in \(\{-1,0,1\}.\)
Problem (D. Grinberg (2020), Institute Mittag–Leffler).
Classify all symmetric functions \(F\) such that, for every partition \(\lambda,\) the product \(\schurS_\lambda \cdot F\) has coefficients in \(\{-1,0,1\}\) when expanded in the Schur basis.
#Multiplication by \(\powerSum_2\)
A 2022 conjecture stated that, for positive \(k,\) the product \(G(k,m)\powerSum_2\) is signed multiplicity-free in the Schur basis. That is, \[G(k,m) \powerSum_2 = \sum_\mu u_\mu \schurS_\mu\] with \(u_\mu \in \{-1,0,1\}.\) This conjecture has now been proved in [Thm. 1.5, CCEF+22].
#Hopf algebra
There is a coproduct formula for the Petrie symmetric functions: \[\Delta( G(k,m) ) = \sum_{j=0}^m G(k,j) \otimes G(k,m-j).\]
#Pieri rule
In [JJL24], the authors give a combinatorial proof of the Pieri-type rule for the expansion \[G(k,m) \schurS_\mu = \sum_\lambda \operatorname{Pet}_k(\lambda,\mu) \schurS_\lambda.\] The coefficients \(\operatorname{Pet}_k(\lambda,\mu)\) are in \(\{-1,0,1\}.\) Y. Jin, N. Jing, and H. Liu give a combinatorial interpretation using ribbon tilings, similar to the Murnaghan–Nakayama rule.
Alternative proofs are given in [WEKS25], and the rule is extended to skew shapes there.
#Generalization
The following definition appears in [Eq. (3.20), DW92].
It is natural to extend the definition of Petrie symmetric functions as follows. Let \[G(k,\lambda) \coloneqq \det\left[ G(k, \lambda_i-i+j) \right]_{1\leq i,j \leq \length(\lambda)}.\] Then \(G(k,\lambda) = \schurS_\lambda\) whenever \(k \gt{} \lambda_1,\) and \(G(k,\lambda)\) are the so-called modular Schur functions.
The \(G(k,\lambda)\) are not Schur-positive in general. For example, \[G(4,44) = \schurS_{44} + \schurS_{332} - \schurS_{431} + \schurS_{611} + 2 \schurS_{2222} - \schurS_{3221} + \schurS_{4211} - \schurS_{5111} - \schurS_{2111111} + \schurS_{11111111}.\] In fact, \(G(5,321)\) is not even monomial-positive.
#Modular Schur functions
The modular Schur functions were introduced by S. Doty and G. Walker [Eq. (3.20), DW92], and further studied by G. Walker [Wal94].
They are indexed by two parameters, a partition \(\lambda\) and a positive integer \(k.\) When \(k\) is prime, the modular Schur function is a character of \(\GL_n(K),\) where \(K\) has characteristic \(k.\)
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