2024-06-04

## Petrie symmetric functions

The *Petrie symmetric functions* is a family of symmetric functions indexed by
two non-negative integers.
This family is the main topic in the work by D. Grinberg [Gri20aGri20b].
They are also studied independently by H. Fu and Z. Mei [FM20]
under the name *truncated homogeneous symmetric functions*.

The earliest reference we have found (so far) is [Eq. (3.11), DW92], under the name modular complete symmetric functions. A corresponding family of modular elementary symmetric functions is also defined.

A more recent paper, [BAB18], studies the
Petrie symmetric functions under the name *generalized elementary symmetric functions*,
when studying variants of $q$-binomial coefficients.
See also the recent reference is M. Ahmira 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 $m,$ where all entries are less than $k.$ Note that for $k=2,$ we have that $G(2,m) =\elementaryE_m.$ Hence, the alternative notation $E^{(k)}_m(\xvec)$ as seen in [BAB18AM20] makes sense — however there is a shift in indexing so that the elementary symmetric functions are recovered at $k=1$ in their notation.

Note that $G(k,m)$ is homogeneous degree $m.$ The non-homogeneous 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 as a special case 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 $a(t)\coloneqq \sum_{t\geq 0} a_i t^i$ 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 clearly 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

\[ \sum_{k=0}^\infty E^{(s)}_k(x_1,\dotsc,x_n) t^k = \prod_{i=1}^n \left( 1 + x_i t + \dotsb + (x_i t)^s \right)^{-1}. \]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:

\[ \sum_{j=0}^k (-1)^j E^{(s)}_j(x_1,\dotsc,x_n) H^{(s)}_{k-j}(x_1,\dotsc,x_n) = \delta_{0,n}. \]There are some interesting relations related to evaluations of monomial symmetric functions at roots of unity given in [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 imply that $H_k[ \schurS_\lambda ]$ is the determinant of a Petrie matrix, which always have value $\pm 1$ or $0,$ see [GW74] for more background.

**Example (Schur expansion of $G(4,8)$).**

We have that $G(4,8)$ is given by

\[ \monomial_{332}+\monomial_{2222}+\monomial_{3221}+\monomial_{3311}+\monomial_{22211}+\monomial_{32111}+\monomial_{221111}+\monomial_{311111}+\monomial_{2111111}+\monomial_{11111111} \]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 [Gri20b]. A more explicit formula using $k$-cores is proved in [CCEFY22]. These formulas are used to prove the Liu–Polo conjecture, [Remark 1.4.5, LP21].

The Petrie symmetric functions have the following nice 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 $\powerSum_2$

In February 2022, I conjectured that for positive $k,$ the product $G(k,m) \powerSum_2$ is signed multiplicity-free in the 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, CCEFY22].

### Hopf algebra

There is a nice 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

\[ G(k,m) \schurS_\mu = \sum_\lambda Pet_k(\lambda,\mu) \schurS_\lambda. \]The $Pet_k(\lambda,\mu)$ are in $\{-1,0,1\}$ and the paper above gives a combinatorial interpretation of these using ribbon tilings (similar to the Murnaghan–Nakayama rule).

### Generalization

The following definition is done in [Eq. (3.20), DW92].

It is natural to extend the definition of Petrie symmetric functions as follows. Let

\[ G(k,\lambda) \coloneqq \det[ G(k, \lambda_i-i+j) ]_{1\leq i,j \leq \length(\lambda)}. \]Then $G(k,\lambda) = \schurS_\lambda$ whenever $k > \lambda_1,$ and $G(k,\lambda)$ are the so called modular Schur functions.

The $G(k,\lambda)$ are not Schur positive in general. For example,

\[ G(4,44) = \schurS_{4 4} + \schurS_{3 3 2} - \schurS_{4 3 1} + \schurS_{6 1 1} + 2 \schurS_{2 2 2 2} - \schurS_{3 2 2 1} + \schurS_{4 2 1 1} - \schurS_{5 1 1 1} - \schurS_{2 1 1 1 1 1 1} + \schurS_{1 1 1 1 1 1 1 1}. \]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 $m.$ When $m$ is a prime number, the modular Schur function is a character of $\GL_n(K)$ where $K$ has characteristic $m.$

## References

- [AM20] Moussa Ahmia and Mircea Merca. A generalization of complete and elementary symmetric functions. arXiv e-prints, 2020.
- [BAB18] Abdelghafour Bazeniar, Moussa Ahmia and Hacène Belbachir. Connection between bisnomial coefficients with their analogs and symmetric functions. Turkish Journal of Mathematics, :807–818, 2018.
- [CCEFY22] Yen-Jen Cheng, Meng-Chien Chou, Sen-Peng Eu, Tung-Shan Fu and Jyun-Cheng Yao. On signed multiplicities of Schur expansions surrounding Petrie symmetric functions. arXiv e-prints, 2022.
- [DW92] Stephen Doty and Grant Walker. Modular symmetric functions and irreducible modular representations of general linear groups. Journal of Pure and Applied Algebra, 82(1):1–26, September 1992.
- [FM20] Houshan Fu and Zhousheng Mei. Truncated homogeneous symmetric functions. arXiv e-prints, 2020.
- [Gri20a] Darij Grinberg. Petrie symmetric functions. arXiv e-prints, 2020.
- [Gri20b] Darij Grinberg. Petrie symmetric functions. Online 2020.
- [GW74] Manfred Gordon and Edward Wilkinson. Determinants of Petrie matrices. Pacific Journal of Mathematics, 51(2):451–453, 1974.
- [JJL24] Emma Yu Jin, Naihuan Jing and Ning Liu. On a Pieri-like rule for the Petrie symmetric functions. arXiv e-prints, 2024.
- [LP21] Linyuan Liu and Patrick Polo. On the cohomology of line bundles over certain flag schemes II. Journal of Combinatorial Theory, Series A, 178:105352, February 2021.
- [Num07] Yasuhide Numata. Pieri's formula for generalized Schur polynomials. Journal of Algebraic Combinatorics, 26(1):27–45, January 2007.
- [Wal94] Grant Walker. Modular schur functions. Transactions of the American Mathematical Society, 346(2):569–604, February 1994.