{"title":"On Antipodes of Immaculate Functions","authors":"John Maxwell Campbell","doi":"10.1007/s00026-022-00632-0","DOIUrl":null,"url":null,"abstract":"<div><p>The immaculate basis of the Hopf algebra <span>\\(\\textsf {NSym}\\)</span> of noncommutative symmetric functions is a Schur-like basis of <span>\\(\\textsf {NSym}\\)</span> that has been applied in many areas in the field of algebraic combinatorics. The problem of determining a cancellation-free formula for the antipode of <span>\\(\\textsf {NSym}\\)</span> evaluated at an arbitrary immaculate function <span>\\( {\\mathfrak {S}}_{\\alpha } \\)</span> remains open, letting <span>\\(\\alpha \\)</span> denote an integer composition. However, for the cases whereby we let <span>\\(\\alpha \\)</span> be a hook or consist of at most two rows, Benedetti and Sagan (J Combin Theory Ser A 148:275–315, 2017) have determined cancellation-free formulas for expanding <span>\\(S({\\mathfrak {S}}_{\\alpha })\\)</span> in the <span>\\({\\mathfrak {S}}\\)</span>-basis. According to a Jacobi–Trudi-like formula for expanding immaculate functions in the ribbon basis that we had previously proved bijectively (Discrete Math 340(7):1716–1726, 2017), by applying the antipode <i>S</i> of <span>\\(\\textsf {NSym}\\)</span> to both sides of this formula, we obtain a cancellation-free formula for expressing <span>\\(S({\\mathfrak {S}}_{(m^{n})})\\)</span> in the <i>R</i>-basis, for an arbitrary rectangle <span>\\((m^{n})\\)</span>. We explore the idea of using this <i>R</i>-expansion, together with sign-reversing involutions, to determine combinatorial interpretations of the <span>\\({\\mathfrak {S}}\\)</span>-coefficients of antipodes of rectangular immaculate functions. We then determine cancellation-free formulas for antipodes of immaculate functions much more generally, using a Jacobi–Trudi-like formula recently introduced by Allen and Mason that generalizes Campbell’s formulas for expanding <span>\\({\\mathfrak {S}}\\)</span>-elements into the <i>R</i>-basis, and we further explore how new families of composition tableaux may be used to obtain combinatorial interpretations for expanding <span>\\(S({\\mathfrak {S}}_{\\alpha })\\)</span> into the <span>\\({\\mathfrak {S}}\\)</span>-basis.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-022-00632-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
The immaculate basis of the Hopf algebra \(\textsf {NSym}\) of noncommutative symmetric functions is a Schur-like basis of \(\textsf {NSym}\) that has been applied in many areas in the field of algebraic combinatorics. The problem of determining a cancellation-free formula for the antipode of \(\textsf {NSym}\) evaluated at an arbitrary immaculate function \( {\mathfrak {S}}_{\alpha } \) remains open, letting \(\alpha \) denote an integer composition. However, for the cases whereby we let \(\alpha \) be a hook or consist of at most two rows, Benedetti and Sagan (J Combin Theory Ser A 148:275–315, 2017) have determined cancellation-free formulas for expanding \(S({\mathfrak {S}}_{\alpha })\) in the \({\mathfrak {S}}\)-basis. According to a Jacobi–Trudi-like formula for expanding immaculate functions in the ribbon basis that we had previously proved bijectively (Discrete Math 340(7):1716–1726, 2017), by applying the antipode S of \(\textsf {NSym}\) to both sides of this formula, we obtain a cancellation-free formula for expressing \(S({\mathfrak {S}}_{(m^{n})})\) in the R-basis, for an arbitrary rectangle \((m^{n})\). We explore the idea of using this R-expansion, together with sign-reversing involutions, to determine combinatorial interpretations of the \({\mathfrak {S}}\)-coefficients of antipodes of rectangular immaculate functions. We then determine cancellation-free formulas for antipodes of immaculate functions much more generally, using a Jacobi–Trudi-like formula recently introduced by Allen and Mason that generalizes Campbell’s formulas for expanding \({\mathfrak {S}}\)-elements into the R-basis, and we further explore how new families of composition tableaux may be used to obtain combinatorial interpretations for expanding \(S({\mathfrak {S}}_{\alpha })\) into the \({\mathfrak {S}}\)-basis.
非对易对称函数的Hopf代数(\textsf{NSym})的完美基是在代数组合学领域的许多领域中应用的类似Schur基。在任意完美函数\({\mathfrak{S}}_{\alpha})上计算的\(\textsf{NSym}\)的对极的无消去公式的确定问题仍然存在,让\(\alpha)表示整数组成。然而,对于我们让\(\alpha\)是一个钩子或最多由两行组成的情况,Benedetti和Sagan(J Combin Theory Ser a 148:275–3152017)已经确定了在\({\mathfrak{S}})-基上展开\(S({\ mathfrak{S}}_{\alpha})\)的无消去公式。根据我们之前双射证明的在带状基上展开无完美函数的Jacobi–Trudi类公式(离散数学340(7):1716–17262017),通过将\(\textsf{NSym}\)的反极S应用于该公式的两侧,我们得到了在R基上表示\(S({\mathfrak{S}}_{(m^{n})})的无消去公式,对于任意矩形\((m^{n})\)。我们探索了使用这种R-展开和符号反转对合来确定矩形无瑕函数对极的\({\mathfrak{S}})-系数的组合解释的想法。然后,我们使用Allen和Mason最近引入的Jacobi–Trudi类公式,更普遍地确定了无完美函数对极的无消去公式,该公式推广了Campbell将\({\mathfrak{s}})-元素扩展到R基的公式,并且我们进一步探索了如何使用新的组合表族来获得将\(S({\mathfrak{S}}_。