{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:43:21Z","timestamp":1772448201675,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We construct for each $\\mu\\vdash n $ a bigraded $S_n$-module $\\mathbf{H}_\\mu$ and conjecture that its Frobenius characteristic $C_{\\mu}(x;q,t)$  yields the Macdonald coefficients $K_{\\lambda\\mu}(q,t)$. To be precise, we conjecture that the expansion of $C_{\\mu}(x;q,t)$  in terms of the Schur basis yields coefficients $C_{\\lambda\\mu}(q,t)$  which are related to the $K_{\\lambda\\mu}(q,t)$ by the identity   $C_{\\lambda\\mu}(q,t)=K_{\\lambda\\mu}(q,1\/t)t^{n(\\mu )}$.  The validity of this would give a representation theoretical setting  for the Macdonald basis $\\{ P_\\mu(x;q,t)\\}_\\mu$ and establish the  Macdonald conjecture that the $K_{\\lambda\\mu}(q,t)$ are polynomials with  positive integer coefficients.  The space $\\mathbf{H}_\\mu$ is defined as the linear span of derivatives of a certain   bihomogeneous polynomial $\\Delta_\\mu(x,y)$ in the variables $x_1,x_2,\\ldots ,x_n$, $y_1,y_2,\\ldots ,y_n$. On the validity of our conjecture $\\mathbf{H}_\\mu$ would necessarily have $n!$ dimension.  We refer to the latter assertion as the $n!$-conjecture.  Several equivalent forms of this conjecture will be discussed here together with some of their consequences.  In particular, we derive that the polynomials $C_{\\lambda\\mu}(q,t)$ have a number of basic properties in common with the coefficients  $\\tilde{K}_{\\lambda\\mu}(q,t)=K_{\\lambda\\mu}(q,1\/t)t^{n(\\mu )}$.  For instance,  we show that $C_{\\lambda\\mu}(0,t)=\\tilde{K}_{\\lambda\\mu}(0,t)$,  $C_{\\lambda\\mu}(q,0)=\\tilde{K}_{\\lambda\\mu}(q,0)$ and show that on the $n!$ conjecture we must also have the equalities  $C_{\\lambda\\mu}(1,t)=\\tilde{K}_{\\lambda\\mu}(1,t)$ and $C_{\\lambda\\mu}(q,1)=\\tilde{K}_{\\lambda\\mu}(q,1)$.  The conjectured equality $C_{\\lambda\\mu}(q,t)=K_{\\lambda\\mu}(q,1\/t)t^{n(\\mu )}$  will be shown here to hold true when $\\lambda$  or $\\mu$ is a hook.  It has also been shown (see [9]) when  $\\mu$ is a $2$-row or $2$-column partition and in [18] when  $\\mu$ is an augmented hook.<\/jats:p>","DOI":"10.37236\/1282","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:51:28Z","timestamp":1578707488000},"source":"Crossref","is-referenced-by-count":28,"title":["Some Natural Bigraded $S_n$-Modules"],"prefix":"10.37236","volume":"3","author":[{"given":"A. M.","family":"Garsia","sequence":"first","affiliation":[]},{"given":"M.","family":"Haiman","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[1996,1,26]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v3i2r24\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v3i2r24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T06:18:07Z","timestamp":1579328287000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v3i2r24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,1,26]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[1996,1,24]]}},"URL":"https:\/\/doi.org\/10.37236\/1282","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,1,26]]},"article-number":"R24"}}